It’s very possible for chlorinated pools to be over-chlorinated, which would result in skin irritation. The redness and itching may actually be mild chemical burns. Usually the eyes are affected the most first, since they maintain a normal pH and chlorine increases it, causing the tissues to swell up and the traditional "swimming pool red-eye".

Another thing to consider is allergies or hyper-sensitive skin. You could be allergic to the chlorine (the same base ingredient in most bleaches, by the way) or your skin could be reacting to even mild amounts.

The best way to tell if you have a sensitivity (which means either allergic to or have a mild reaction to) to chlorine is to see if other people have similar blotches. If not, then chances are you have a sensitivity to it and should see a doctor for relief and a note to stay out of the pool. On the other hand, if others are experiencing the same thing, then the chlorine in the pool is too high and the maintenance people need to be made aware that they’re frying their swimmers.
References :
pH Explained: http://en.wikipedia.org/wiki/PH
Chlorine allergies explained: http://www.helium.com/items/1198980-chlorine-allergy-allergies-sensitivity-chemical-sensitivity-pool