How can you tell if a rash is caused by an allergic reaction or something else?

Can't. There's an old saying in dermatology: you can't diagnosis a rash over the phone. I even get confused after seeing some rashes. Generally, allergic rashes are welts but some allergic rashes are small red dots. Best to see your dermatologist.
Rash. You really need to have a doctor look at it to determine the difference. Allergic rashes tend to be short-lived and might be a simple hive appearance to it.