I might ask for this skin test just to rule out coeliacs, but my GP is stingy with tests.
Have you ever been tested for allergies? A wheat or barley allergy will cause the same problems, but will rarely progress to Celiac Disease.
If you don't think your GP will do a test (I don't know how expensive the skin test is, and like I said, it's not always conclusive), you can always just try a gluten-free diet for a month and see if you find any relief from your eczema.
Bear in mind, though, that you have to be COMPLETELY gluten-free - there is no such thing as "gluten lite".
For me, once I was off of gluten I felt SOOOO much better - and every time that I accidentally ate it, I started feeling just awful. Most people who have gluten sensitivity who go on a GF diet find it VERY easy to stick with the diet for that very reason. My sister-in-law and nephew (only related through marriage - we don't share any genes) were diagnosed with CD around the same time - completely unbeknownst to me - and they have said the same thing. It's almost like a "miracle".
If you decide to go GF, commit to doing it for at least a month - if you're not gluten-sensitive, you won't feel a difference (but it can take a couple of weeks to get it out of your system).
I always tell people that a GF "diet" is much cheaper than all of the doctors' visits and medicines I was paying for before. I still have MG, but there are a host of other conditions that I am less likely to develop as a result of an untreated gluten issue.
FWIW, there is a MUCH higher incidence of CD in Ireland - they don't know why. ALL of my ancestors were from England, Ireland, or Scotland, so it's not surprising that I have this....