my son had eczema when he was few months old too, so it was quite similar to your case.
and I switched his milk formula to 低敏, soy, or goat milk, still the same.
so when he turned 1yr old, I brought him to do skin prick test.
and his allergy specialist recomended another formula called Neocate to him, which is an amino acid base formula. and he drinks it now still.
his skin has really big improvement once he changed to Neocate. but this formula is quite expensive and you have to purchase directly from their distributor.