I'm really not sure what you mean by dead skin. Perhaps you mean dry and rough skin, which would suggest that what you have is eczema. You might want to try hydrocortisone 1% (OTC) twice daily--seeing whether that makes the problem go right away would help clarify the situation. However, for definitive diagnosis and treatment, you'll need to see a doctor.
This does not sound serious or infectious, however.
Best.
Dr. Rockoff