to which your child is allergic by getting allergy testing done from an allergist.Also wash the areas several times with fresh water. Do not use any cosmetic products at the sites. You can apply some calamine lotion at the rash as it will help in soothing the skin. You can give some OTC oral antihistamine medications like Benadryl or Claritin and see if it helps.For mild-moderate eczema a weak steroid may be used (e.g. hydrocortisone as dermacort), whilst more severe cases require a higher-potency steroid (e.g. clobetasol propionate, fluocinonide).But these should be taken only after prescription from a dermatologist.
Eczema can be exacerbated by dryness of the skin. Use good quality moisturizers/emollients to prevent moisture loss from the skin.
Hope it helps.Take care and pls do keep me posted in case you have any additional doubts.