It is hard to identify a rash without seeing it. According to the description you provided you might be experiencing a kind of allergic reaction, possibly eczema. Eczema is a form of skin inflammation characterized with redness, skin edema, drying, and itching. However, in certain subtypes of eczema, itching is not so easily detected.
You might want to try different moisturizing products to decrease the dryness and ask your doctor about corticosteroid anti-inflammatory or over-the-counter creams to reduce the inflammation.
If this does not help within several weeks of application, then you might want to visit a dermatologist.
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