Can the skin become irritated from exposure to chlorine or other swimming pool chemicals?
Yes, this is possible, although it is not common and there are lots of other things that can irritate the skin. The chlorine can dry out the skin, making it irritated. The chemically treated water also could cause contact dermatitis, a rash, which occurs when the skin comes into contact with an irritant or allergen.
Just because it is possible that the pool chemicals are irritating your skin, do not assume that is what is going on.
It often takes quite a bit of sleuthing to identify the cause of contact dermatitis, because our skin comes in contact with so many different substances. Some sunscreens, for instance, can cause contact dermatitis. Plus, the skin can suddenly react to something it has been exposed to many times before without any problem.
It probably makes sense to start your detective work by making sure the pool water is not being overdosed with chemicals, that everything is in the right balance and that all the pool treatment systems are working properly.
Talk with a doctor or dermatologist that can help you devise ways to identify the culprit and recommend treatment for your irritated skin.