Given a rotation $R$ in $\Bbb R^n$, how many pairs $(x, R(x))$ would you need to know to characterize the rotation.
For a general linear transformation you need $n$ such pairs. That is, you can completely characterize a linear transformation $T: \Bbb R^2 \to \Bbb R^2$ by knowing how two linearly independent vectors, $b_1$ and $b_2$, are transformed by the transformation, $T(b_1)$ and $T(b_2)$.
A rotation is not just a general linear transformation though -- it has some unique properties. For instance $R \circ R^* = Id$ (where $R^*$ is the adjoint transformation). I would imagine that these properties would add extra constraints to the problem, probably allowing us to use a smaller set of pairs to characterize the rotation.
So how many pairs $(x,R(x))$ do we need to know to characterize a rotation $R$ in $\Bbb R^n$?
A rotation around the origin is completely identified by the assignment of a nonzero vector $\vec n$ and and angle $\theta$. The vector $\vec n$ gives the axis around which to rotate $\theta$ radians.