Reply to headhunter

I wouldn't use "skip"; it's highly informal.
Try,

That is truly a very attractive opportunity, but I am afraid I have to decline that offer.

Thanks! However, the headhunter only told me that there was an opportunity to have interview with the company. The company did not give me any offer. Could I say "That is truly a very attractive opportunity, but I am afraid I have to pass."?

Again, "pass" would be informal.
Try,
That is truly a very attractive opportunity, but I am afraid I have to decline that offer for the interview."

As it will only be a casual conversation between me and the headhunter, would the words skip and pass be acceptable?

Why don't you want to go to that interview?

Various reasons which I do not want to tell the headhunter.

I'm not sure if I should say between me and the headhunter. Should it be between I and the headhunter?

Plusluck As it will only be a casual conversation between me and the headhunter, would the words skip and pass be acceptable?

I still wouldn't use either of those.
Try,
That is truly a very attractive opportunity, but I am afraid I won't go for it."

Your fundamental problem is that most of your sentence is formal but you are adding an informal expression at the end..

If you want to write the first part informally, say
eg Thanks for the great opportunity, but . . . .

Thank you all. If I want to give a reason to the headhunter, could I say "it would be more attractive to me if the vacancy is from an international company"?