Hi Chilli-
I get the same rash around pretty much any dog, as well as in dusty areas, around cats, when things are blooming, etc. The thing with my Basenji is that I haven't had any respiratory allergies to his dander, and there are some things you can do to prevent/manage the rash:
Make sure you are washing with something like Dove or Aveeno, a good moisturizing soap. Soaps like Irish Spring will take the protective layer of oil off of your skin and leave it susceptible to allergens, I learned that from my dermatologist.
Moisturize every day. Use a good moisturizer (again) like Dove or Aveeno.
After cuddling with the pup, wash your arms and hands. (sounds easy- but it is easy to forget as well)
Try to keep from scratching, depending on your skin, itching it can make hives appear (at least it happens to me occasionally). I find that slapping it can make the itch stop for a while (learned one when I got my first tattoo)
If at all possible, use a groomer to bathe your dog…doing it yourself can expose you to a lot of the dander.
If needed, take some benedryl, or another antihistamine
After a while, your body will get used to your dog's specific dander, and you will have less issues.
I'm not a doctor, and don't know if these solutions are medically correct, this is just what I have noticed, having had the same problem. I have been able to manage my rash and hives using these ideas.