[Note to readers: This article makes a distinction between “man” and “male”, using “male” to refer to biological anatomy and “man” to refer to qualities of character and personality. It is a clumsy way of making the distinction, for which I apologise, but I couldn't think of a better one.]
A man is not defined by his relationship with a set of arbitrary and fashionable qualities. A man is defined by his relationship with God. The more a man submits to God the more manly he will become, however this fact is not well understood because many people don't understand the nature of God and, indeed, many people go to some trouble to deliberately misunderstand the nature of God.
God can be compassionate, merciful, creative, imaginative, magnanimous, rational, tender hearted, sorrowful, awesome, joyful, bold, fearsome, fearless, a mighty warrior, a tender lover, a protective parent, an unmoveable judge, a gentle intercessor. In short, God expresses every good quality and has conferred the potential for every good quality onto men because mankind is created in the likeness of God. A person of male gender is a man only to the extent that he is able to express these same qualities when they are appropriate.
Consequently as a male person submits himself to God he becomes more of the man that God intended him to be. Thus a true man can be aggressive when aggression is rightly called for, merciful when mercy is rightly called for, steadfast when steadfastness is rightly called for, resolute when resoluteness is rightly called for, creative when creativity is rightly called for, just when justice is rightly called for, courageous when courage is rightly called for, passionate when passion is rightly called for, fearsome (i.e., inspiring fear) when fearsomeness is rightly called for, gentle when gentleness is called for, ... and so on. The true man is always honest to the fullest extent of every sense of that word.
A woman's desire will be for her man and he will rule over her. But the man referred to in that passage in the book of Genesis is not the person of male gender who populates the corridors of power or flaunts himself in our media or who bumbles indecisively or measures his progress through life by the numbers on his bank statement. Our fascination with “heroes” is not just some transient sociological phenomena but is because those so-called heroes do, in some small and incomplete way, resemble man as God intended man to be; as God designed and created man to be. Women were created and designed to respond to that kind of a man; they were created to be his helper, to be the one who is capable of both nourishing his strength and also of being nourished by it. They were created not to be his equal but to be his complement, in the sense of being that which makes a thing complete, makes it perfect and brings it to peak performance. In that role they can be satisfied.
The man of God is not limited to being a naive philanthropist wrapped in sackcloth and sporting a haircut that even his elderly neighbour's decrepit pet dog would be embarrassed by. The man of God can be like David (the former King of Israel) who was in turns the ferocious warrior, wise ruler, passionate lover, magnanimous and merciful victor who remembered to say his prayers like a little child and who repented humbly even after committing one of the most appalling acts of sin recorded in the whole bible (namely, the cowardly and treacherous murder of his loyal servant). The man of God can be like Paul (known as an Apostle), who stayed firm under severe pressure who remained cheerful under torment, repented of his persecution of Christians, remained faithful in the face of rejection and, in between penning missives full of concern and care, ran around the world having the sort of splendid adventure that wouldn't look out of place as a chapter in a tale of Sinbad the sailor. The man of God says his prayers on his knees and puts his trust in God but that doesn't stop him from using his initiative and working boldy and energetically to achieve something worthwhile.
Let's forget all that boring twaddle about a woman wanting a man because she perceives him to be a good source of chromosomes for her potential offspring ... a woman wants a hero because heroes are exciting and interesting, they are enlivening and they are sometimes a lot of fun to be with. The man of God is a hero to the need of the moment and a woman wants to be ruled by such a man because that is what she was created to be satisfied by; in his presence she becomes more intensely alive emotionally, intellectually, erotically, creatively. Confronted with such a man she might even be able to overlook the dodgy haircut (it can be dealt with later after all).
However for the lonely woman seeking this kind of hero—the man who will take her in hand and contribute a new vitality to her life just as she encourages him in his—there is a problem: Sometimes it seems that our society is determined to prevent persons of male gender from becoming the men that God intended them to be, and does so by refusing to contemplate or teach the necessary qualities, by ridiculing those qualities and by compartmentalising them. Furthermore our society tinkers with truth, dabbles with deceit and regards morality as a fashion and not, as it actually is, an unchanging quality of the universe which is itself defined by, and is an expression of, the nature of God. Those persons of male gender who wish to become men must first work out what a man actually is because society certainly will not teach them. Many male persons either abandon the quest at a point of pleasure or simply don't live long enough to make the discovery.
How can a woman be satisfied when the thing she herself is designed to be satisfied by, namely a man, is in short supply? That makes life a little difficult so, in place of a man, she might substitute something else; usually something that bears a passing resemblance to the shadow of a man. It might be an aggressive male person, a pushy male person, an assertive male person, a clever male person, an ambitious male person, a strong male person, or a caring male person. Some don't even aim that high and just choose the male person with the most expensive looking car, hoping that expensive cars and manliness are somehow related. If a woman is lucky she will find a male person who has enough of the right qualities to make her respond as she was created to respond: kneeling, adoring, supporting, encouraging, strengthening him by yielding herself, giving her mind, her energy, her attention, her love and her passion ... and receiving back the same from him. However many women (perhaps most) are not so lucky and, in any case, many of them are just as confused as the males they associate with and would rather accept something less challenging, less demanding. Of course some didn't want a man at all but just wanted the expensive car with or without the male accessory pack.
Many women feel a deep need for a man to take charge, “wear the trousers”, or as this forum describes it “to be taken in hand”, because they were designed to yield to a man as I have tried to describe him. It is not a educational or cultural issue but a creation issue. They desire to yield to such a man for the same reason that they desire to fill their lungs with clean sweet air: Their lungs were created to breathe clean sweet air and they were created to yield to a man. However just as the air in our cities is not the type of air women were designed to find satisfaction in breathing, neither are the men in our cities the type of men that women are designed find satisfaction in yielding to.