Yes if bleach does come into contact with your skin it can burn your skin and cause irritation.
When handling bleach you should always wear disposable gloves to keep the bleach off your skin when using it to clean with.
If the bleach is left on your skin too long it can cause permanent tissue damage so always wash the bleach off your skin right away if you do get bleach on your skin.
Avoid getting the bleach in your eyes as well because the bleach in your eyes can lead to permanent blindness and if you do get bleach in your eyes you need to wash the bleach out of your eyes using water immediately.