You might have missed this earlier in the series, but Elaida has clearly shown and stated beliefs that indicate that she doesn't trust most men since she values her fellow Red Ajah Sisters much more but her hatred is reserved for darkfriends - she is one of the Sisters that happens to think that male channelers are darkfriends for channeling and not coming immediately to the Tower to be gentled, which is a pretty typical view for a Red Ajah Sister.

There is a great article written on just this subject by Linda Sedai (see link below), and it essentially keeps track of Elaida and other Aes Sedais' statements regarding Ashaman and male channelers. There are many indications of her opinion, but I think the overall point is that she is one of those people who is not a moderate (definitely not moderate enough to be a Gray nor independent enough to be a Blue) and tends towards extremism. This is why she's called a hard woman, and if you notice, most of the major leaders of the Red Ajah share this view. Keep in mind that this Ajah was modeled on the Inquisition, and it is easy to see why their senior leaders would be like this. Further, the few Red Sisters with moderate views tend to be in weaker positions or poor favor - Pevara is in low regard because she suggested that Warders (i.e. men) would be able assistants while Tarna, as an example of a moderate in a pretty good position, has extreme hatred towards Katerine - there is usually solidarity between fellow Ajah members, but the way RJ wrote the character, we are meant to think that Tarna has such an extreme reaction to Katerine because she detects something evil about the other woman to the point where she can't hide her opinion.

Anyway, Elaida is an extremist in her Ajah, but she is not really a good fit for any of the others, so it seems to make sense to me that she ended up in the Red. Especially since she likes power and the Red Ajah has the most clout and members.