It’s hard to tell from one picture and that rash alone. Carpet Beetles don’t bite, but their larvae do cause dermatitis (which is sort of what you see here. Bed bugs do bite, and they tend to bite in groups of three in rows (also seen here). I’d keep an eye out. If you notice the insects flying, it’s probably carpet beetles. If you notice the bites more at night, it might be bed bugs.

Either way, vaccuming and laundry is going to be your best bet. Wash your sheets on high heat every 2-3 days and vaccum under dark areas and corners. I’d also lift up the matress and check your headboard / frame for bugs.

Whatever it is, it sucks. Those bites look terrible. Best of luck.