Fish oils are rich in omega-3 which is essential for healthy skin. Omega-3 food and supplement sources can help alleviate the dry skin associated with eczema. However, adverse reactions have been reported when using a fish oil supplements to treat eczema, and it could prevent your eczema from clearing up.
Fish oils are naturally rich in amines, which can cause a worsening of eczema symptoms in 30% of eczema sufferers according to research by the RPA Hospital allergy unit. How do you know if you are one of the 30% who should avoid amines? It's usually only possible to tell via an elimination diet (the one I recommend is The Eczema Diet).
If you have eczema it's best to avoid fish oil supplements while following a treatment plan.
Omega-3 rich alternatives:
whole linseeds (freshly ground)
flaxseed oil (fresh only, refrigerated, use within 5 weeks of opening)
Fresh white fish (most other fish contain amines, see next)
Note that frozen fish or leftover fish (any fish or meats eating the next day after cooking) are rich in amines, and if you are sensitive to amines you may react to salmon.