Why are women allowed to wear hats indoors, but not men?

Hat etiquette is not that universal! In Western (Christian-based) cultures, removing headgear is a sign of respect and obedience for men; covering the hair is a sign of modesty in women (and leaving a fashion hat on is also practical, as it may be difficult to remove). This distinction, however, is not usually considered to apply in the case of unisex hats (e.g., baseball caps should be removed by members of both genders). Other cultural traditions have different 'rules' regarding hats.

https://en.wikipedia.org/wiki/Headgear#Headgear_etiquette

Interesting, thanks for that info looney_tunes.