Yes, they make a round cover (about 5" dia.) for this. You should be able to find it beside the switch and receptacle coverplates at the hardware store.
If the area you want to cover is larger, an electrical supply store would have larger medallions that can be used in conjunction with the 5" cover.

Yes, and they're pretty close to flush (but not perfect). Just paint to match the ceiling, and most people won't notice it.

Various size covers/medallions are in the fan section in the big box stores.

Thanks all!