Chlorine reacts with your skin. Here are two videos showing the effect.
This video shows how quickly chlorine from tap water reacts with a person's body (here the mouth). They show that tap water contains chlorine by using an indicator (yellow means there's chlorine in the water). Then the subject swishes the chlorinated water with his mouth for a few seconds. That water is then spit out and shown to have no chlorine (the indicator is clear, meaning no chlorine). The author reasons that the chlorine was absorbed by the mouth because it's no longer in the water. This video shows a similar effect using chlorinated water and the skin on person's hand. Both videos show that chlorine in tap water reacts with the biological fibers making up a person's hair and skin.
If chlorine in tap water reacts with your skin, what about the pool? There, your entire body is exposed to pool water which is has more chlorine than tap water. Your body reacts with lots of chlorine. You become a new "chlorinated you." Accordingly, many swimmers notice that they "reek of chlorine" after swimming because their bodies are saturated with chlorine. The next question should be: is chlorine bad for you?