Just to add a little note to what Alan said (which is absolutely correct), when I teach rotation, I find the easiest way to grasp it is to think about how to most easily describe the rotation of a rotating object. Clearly if it's rotating about an axis, we can pin down that axis. Then what remains it to describe *in what sense* it rotates about that axis, *i.e.* clockwise or counter-clockwise as viewed from a particular reference frame. We can define this by coming up with a rule, and we choose the "right hand rule" just so that we all agree on notation. So we put an arrow head on our rotation axis and say that the direction of this axis specifies rotation of the object via the right hand rule.

So in summary, we've defined a rotation axis and used the right hand rule to specify in what sense the object is rotation about the axis, which is sufficient to describe rotation about that axis.