To prove that a matrix is a rotation matrix, you have to check :
- vectors are norm 1 (lines or columns, it's not important)
- determinant > 0 (since vectors are norm 1, it automatically implies that the determinant is =1)
The axis will be the eigenspace generated by 1 (I'm not sure if there are further things to do here) :
It's like in geometry : the axis doesn't change when rotating.