Definitely hats off indoors (and in cars, too, if you ask me).
I don’t wear hats often but, during a trip to the Christmas Markets in Germany, a couple of years ago, the frigid conditions more or less dictated the use of a nice, cosy Thinsulate wooly hat.
Mrs beluga decide that she wanted to visit Cologne Cathedral, so in we went. I’d completely forgotten about the hat and, within about twenty feet, was approached by an usher, who gestured for me to take the hat off. I obliged, and carried on through to the main part of the Cathedral, where I was confronted by lots of ‘Ladies of a certain age’ – most of whom were wearing hats!
That’ll teach me to be a member of the least defended minority – a white, middle aged, straight, agnostic man 😉