At the end of a letter, is it correct to omit "I am" in this sentence?
I am looking forward to hearing from you
Then it will become:
Looking forward to hearing from you
“Looking forward to hearing from you” is a written representation of Conversational Deletion, which John Lawler has treated on ELU: here, for instance, and here. This term names the very common colloquial omission of anything at the front of a sentence which can in context be taken for granted.
Because this is a colloquial practice, it should be used only in letters to people with whom you are on familiar terms: friends, family, or business and professional associates with whom you have a long and friendly history. Do not use it with correspondents whom you do not know well. Use the full form instead; or for just a touch more formality use the simple present:
I look forward to hearing from you.