Which one is correct?

They are both fine.

You only ever add (s) on to the end of "skill" when you DO NOT include the word "a" in the sentence:
e.g. You have very good cooking skills.
e.g. You have a very good cooking skill.

But you would usually say, "you have very good cooking skills", because there are a lot of different skills in cooking (and obviously for any other type of word, e.g. writing skills, catching skills), so therefore it is a plural.

his writing skills are excellent.

Your first sentence should read as followed..... His writing skills are excellent.
The 2nd sentence is fine the way it is. Good job.
People usually say "skills".