Funny thing about that moon. I found out recently that it's been speculated that the reason the moon looks so angry isn't because it's an evil moon that's deliberately trying to destroy the world, but that it's angry because it's being forced down to destroy the world against its will, struggling and straining to fight back and resist the downward pull of the mask's magic being forced on it, meaning it's also another victim of the Majora's Mask.
This idea is based on the fact that the mayor of Clock Town has an ornament on the back of his office chair that looks like the very same moon but with a smiling happy expression on it, and based on certain comments of the townspeople saying how the moon's appearance looks unusually different than normal. While the latter could just be taken in regards to the fact that it has a face at all, at no point in the game does any NPC remark on that particular notion as being abnormal. If anything, the mayor's chair depicting the same moon with a happy face implies that it's supposed to have a face, and that what's different about it is its current expression of gritted-teeth anger.
With this idea in mind, looking up at that moon puts that angry face in a whole new perspective, one of sympathy and tragedy instead of outright fear and anxiety.