Hi firstly there can be many reasons why one will have eczema. It may be because allergic to dust, certain foods, or even genetic. My niece has eczema and she went to Kkh to do allergy test.
http://www.kkh.com.sg/services/children/allergyservice/Pages/Home.aspx
She did a skin patch test and conclusion is she is allergic to dust mite. Maybe you can consider bringing baby there for a test? Switching milk may help , you can trial and error to see if that helps too.
Jinhui Fu Jacinthia