Hello,
The symptoms are suggestive of tinea versicolor. Contact dermatitis or allergic reaction to any detergent will not cause any white dots but red, itchy rashes. Tinea versicolor is caused by a fungus that lives in the skin of almost all adults. Factors that can cause the fungus to become more visible include high humidity and immune or hormone abnormalities. Treatment involves topical and oral antifungal medicines. If the symptoms persist then a dermatologist’s evaluation will be the best.
I hope it helps. Take care and please do keep me posted in case you have any additional doubts. Kind regards.