My father wasn't too strict on manners (or maybe I got them instilled quickly), but whenever I left an elevator before a woman (even I was closer to the exit), he would grab me by the collar and pull me back.
To this day, if I exit an elevator before a woman, even if it's the only way she can get out, I get a choking sensation.
Don't know this from personal experience, but I have read that if a woman tries to exit an elevator first in Japan, she risks being trampled by the men who expect her to wait till after them. Also that women drivers yield to men srivers at stop signs. Interesting, no?