I'd recommend going to see your primary care doctor
about this issue. There are so many different causes of the symptoms you are describing, and it is really difficult to narrow down this list of causes without a close in-person inspection of the rash.
You are definitely right that an allergic reaction of the skin is high on the list of possibilities. This could be caused by exposure to any number of substances, such as new personal care products, laundry detergents, soaps, etc.
It is also possible that this could be eczema
, caused by drying out of the skin now that we are entering the cold winter months of the year.
Another unrelated possibility is that these red spots may represent bug bites. For example, flea bites (if you have pets), bed bugs (from infestation of a mattress or bedding), or scabies can all cause intensely itching red spots on various areas of the body.
Your doctor should be able to make a diagnosis after taking a look at the spots. Treatment will obviously depend a lot on what the diagnosis is, but should include at least some topical medications to reduce itching and inflammation. Book an appointment with your doctor today!