Why would evolution select for greater special awareness in men but not in women? Our ancestors swung through the same trees - one sex with children on their backs.
Swinging through the trees, that's going back a ways!
There are other species that have differences in these sorts of things, so I don't see it as impossible. But it might not to relate to the kind of selection you are thinking of, but traits aren't always about a survival advantage. It could be something less direct - maybe it really relates to females superior language skills, there is some kind of trade-off involved.
Just to compare, if you asked why evolution would select for women to be slower than men, since they also need to run away from tigers, you'd be right to think that being fast would be just as useful for women. But - women also have to have the physiology to have babies, so, they are slower. A trait that differs by sex, but not due to evolutionary pressures around speed.