Regarding eczema in an infant...My daughter had eczema that was strictly related to food allergies . I discovered while I was nursing her that she was allergic to corn, soy, milk, eggs and a whole bunch of other foods. I thought that Aunty's suggestions were excellent and very worthwile. My daughter's skin only improved when we removed the offending foods from my diet. While prescription creams helped, her skin cleared up immediately when we got to the root of the problem.
Pinpointing specific food allergens in an infant or very young child is difficult to say the least but it can be done and done with precision. After spending 20 months suffering through it alone, we ended up finding help with our daughter through a very specialized allergist (as in there are only a few like this in the U.S). Rule out as much as you can by eliminating possible causes. Have you considered using only baking soda to wash your child's clothes and everything that her body comes into contact with? That has been tried and proven by another family with a child suffering from extreme eczema. Dust, mold and other environmental factors can also be related to eczema. Hang in there, I know how upsetting your situation may be as it was not that long ago I was in a similar space.