One must remember that everything we see and touch except for light and electricity is a 'chemical' -- even a newborn babies skin is 100% chemical. Therefore, when we choose to use household cleaning solutions without wearing protective gloves, we expose ourselves to the skin damaging effects of cleaners. Most household cleaning solutions will damage the surface layer of skin, making it peel off in layers or in chunks, though not necessarily the entire skin surface is compromised. You could use an exfoliating cleanser to help alleviate the peeling skin and follow with a mixture of jojoba oil and vitamin E oil to help condition, moisturize and heal the damaged skin.
When I clean my house, I wear surgical gloves for protection against the harsh chemicals. Wearing surgical gloves, as opposed to those huge, thick plastic ones, allows me to 'feel' what I am doing/touching while protecting my skin. Also, allergic skin reaction developes from repeated and prolonged exposure to a known skin sensitizer, and ALL household cleaning solutions fall into this category.