When you use the right hand rule and actually try to do that with your hand, you almost always get a non-orthonormal basis. If the angles between you fingers are very oblique, that is a good example; non of the angles may be zero, however.
More formally, it should be any three vectors not pairwise collinear, and forming a right-hand triad.
It basically boils down to a positive determinant of the all the basis vectors in their natural order. If it's positive then it's right handed, if it's negative it's left handed (you can't get zero and if you do it means you've done something wrong).