My baby brother has severe eczema, all over his body, just everywhere, poor little man. He's eight now, and while mum's never been able to make it go away completely they've had the absolute best results by focusing on diet. He now only has a small patch on his lower back, which has never gone completely, and occasionally his skin will rough up if his diet has been too "relaxed".
She also does the natural detergents, etc. as well, but for sure his diet has worked the best.
She tried all the creams, but they never helped him very much.
She took him to a doctor, and they did an allergy test on him and he now has a list as long as his arm of food to avoid.
At first mum was very strict about it, but after a few months she has relaxed a lot/doesn't make a big deal if he wants to eat someone on his "no" list and because he really likes not having bad eczema he mostly polices himself, with no big hassles made of it. But like I said, he's 8 so that bit wouldnt apply to your baby yet.
The list included things like tomatoes, preservatives, pineapple, vitamin C (as an additive), lots of things.
I'm sure you'd have to get it personalized.
You didn't mention if you breastfeed or not...if you do I think the diet would apply to you??
If you formula feed, it could be the formula, as that would have stacks of preservatives in it I think.
But yea, ask for an allergy test?