OK, dudes are supposed to have your head covered in a Synagog. Fair dinkum. But why a yarmulka or a scullcap instead of a fedora or even a Stetson?
Probably because hats are more distracting. I have seen people opt for a hat instead, though. Also, women tend to wear hats if any head covering while in synogague. (I suppose this is purely for fashion purposes; those little pieces of cloth and yarmulkes are less attractive than a nice hat.
So would a nice fedora be proper?
Any hat would be fine. You can think of a yarmulka as the minimum requirement. A ballcap would be “legal” but most people would find it inappropriate.
Haj
Jewish males are actually required to cover their heads at all times, not just in a synagogue. And hats are perfectly acceptable and are sometimes used, but skullcaps are much smaller and easier to wear inside (and aren’t as hot).
Non religious Jews sometimes skip this law, but wear them out of tradition while at a religious service.
From Aish
by Rabbi Shraga Simmons ;j
From Ahavat-Israel
From Sichos
In addition: Non-Jews may wear them as a sign of respect and understanding if they’re at a Jewish wedding or funeral. You don’t have to talk the talk, but it’s appreciated if you walk the walk. I include among my prized possesions the few yarmulkes from my friends’ weddings. If you’re going in, grab a beanie. It’s the respectful thing to do.
One of our Jewish pals stood up at our wedding in an Episcopal church and presented the bible reading. We gave him the Old-Testament one. 
We asked the priest beforehand if that was ok (kosher?), and he had no complaints. Said something like “my boss was a Jew, I don’t suppose he’d mind.”
For some reason, I broke out laughing when I read that, thinking about a service where those in synogague are wearing baseball caps and trucker’s hats.
I’ve seen people with all sorts of hats and caps in shul.
Zev Steinhardt