This question is closely related to angle and plane of rotation in 3D geometric algebra. For example, Doran & Lasenby already showed a proof for the 3D case, but things are a bit different in 4D.
In general, a rotor in 4D consists of a scalar, 6 bivectors and one four-vector (in 3D, a rotor is just composed of a scalar and 3 bivectors). Then, assume that we already know two 4-dimensional vectors and one is a rotated version of the other. How is it possible to derive a similar expression for the rotor? does it exist? If so, could a general expression for n dimensions be obtained?