Can someone else write this? Not on your life! (13)
I believe the answer is:
autobiography
'can someone else write this? not' is the definition.
The definition suggests a singular noun which matches the answer.
'your life' is the wordplay.
'your' becomes 'auto' (I can't explain this - if you can you should give a lot more credence to this answer).
'life' becomes 'biography' (synonyms).
'auto'+'biography'='AUTOBIOGRAPHY'
'on' is the link.
(Other definitions for autobiography that I've seen before include "Person's life story written by the subject" , "Story of one's life" , "A person's account of their life" , "Personal account of one's life" , "Literary genre" .)