The different traditions for men and women regarding hats have lingered and in many cases there are no reasons for there to be any difference. A man and a woman, both with baseball caps simply sitting on their heads, should either both be able to leave them on or both be expected to take them off indoors. However, there are still some women who wear more elaborate hats as part of their outfits which are actually attached to their hair using hairpins. The hat essentially becomes a part of their hairstyle. Removing the hat would either destroy their hairstyle or it would at the least be very difficult for them to replace the hat without a mirror and/or assistance. In cases like that, it make sense that they would simply keep the hat on. Personally I don't think it should be a double standard based on gender at all - I think anyone whose hat must be affixed to their hair with pins should be permitted to keep it on indoors or during the National Anthem. It shouldn't matter if they are male or female. Actually, even some etiquette experts agree with this - Miss Manners has said that the different rules should apply to the different types of hats regardless of the gender of those wearing them, and that if women choose to wear more casual hats such as baseball caps then they should be expected to follow the same rules that apply to the men who wear those hats. Presumably, then, men who wear dressier hats which require pins could keep them on indoors without violating etiquette rules at all. I do suspect their aren't many males who choose to wear those sorts of hats, though.