This is close to what my arms looked like when I was in high school. Doctor diagnosed the rash as Contact Dermatitis. Maybe. Maybe it was "Contact Eczema". When I had a kidney stone, my mother blamed the shots I was given for the rash, saying she read, well, whatever she read, I have not thought on this subject in many years and my once common knowledge is lost in memory land.
Oh how it itched and I scratched. I had to wear what I remember as gauzy brown sleeves to cover the rash. They were not helpful; slid off as I slept and when the itching got so bad, I could not stand it another second, I scratched my arms raw. Do not remember it spreading to my hands. Doc said I was allergic to soap. Hooray, no more washing dishes! He told her to use Dial bath soap.
When I washed dishes, I leaned my arms on the edge of the sink; perhaps because I was short, dunno, and only vaguely recall soapy water dripping down my arms. But, my mother used Ivory soap (so pure it floats), Ivory laundry detergent as well as Ivory dish detergent. Do not know how long I got out of washing dishes. Think the rash went away only to recur.
After I left home, it never returned. I did have what a sister calls "housewife hands" ~ another intolerably itchy rash on the hands. That was not as bad as my high school arm rash, yet just as annoying.
For years, filling out Medical forms, I answered the allergy question: Soap. Actually, I learned I was allergic to sodium lauryl sulfate.