Yes you can and should cut off dead skin from a wound.
The cutting off of dead skin or removal of dead skin from a wound is known as debridement.
Debridement of a wound helps to promote the healing of the wound so the wound will heal much faster than it would if you didn't cut off the dead skin.
You don't have to cut off the dead skin from the wound to help it heal but doing so does help wounds heal much faster.
You can also and should put some hydrogen peroxide on the wound to help prevent infection as well.