No, you are not asleep. What can happen though is that you could potentially fall asleep due to relaxation. In this situation, while you would still be potentially subject to suggestions, the hypnosis would be much less effective, if not entirely ineffective. Don't worry, though, as the hypnotist regularly makes sure that you are not asleep.
The reason why people are usually asked to close their eyes is mainly to be focused and not be bothered by their surroundings.