https://en.wikipedia.org/wiki/Rotation_matrix would be a great place to start to get a good idea of how you can use a rotation matrix to rotate a vector in 2 or 3 dimensions.
https://stackoverflow.com/questions/22099490/calculate-vector-after-rotating-it-towards-another-by-angle-%CE%B8-in-3d-space This post seems to be roughly what you're looking for.
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To specify a rotation, you need both the axis of rotation, and the angle. Just the angle alone is ill-defined.
Even in 3D space, the number of vectors in a given angle to another vector are infinite, so which would you pick in an n dimensional space? In general calculating the angle between two vectors is a loss-introducing function (you take two sets of n numbers and condense it down to one). You can narrow it down to a set of N-1 linearly independent vectors that form a base of the given set of vectors in a given angle. Somebody with more linear algebra knowledge feel free to correct me.
/r/learnmachinelearning
That's basic linear algebra