How an N dimensional rotation matrix can be constructed with more than one degree of freedom

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Say we have a point or a line in a three dimensional space, which we can rotate it around the origin by two angles, one with respect to axis 1 and the other with axis 2 or 3 (two angles(degrees of freedom) available to rotate this point around the origin.

I basically look for a general formula to construct a rotation matrix in n dimensions while it makes sense to me to have n-1 angels given as input, like the 3 dimensional case.

The formulations that I have found for n-dimensional rotation matrices have only a single degree of freedom (angle) to rotate over it. For example in this quesion which obtains its formulation from here.

Do you know a general formulation of what I describe, for example in a 3 dimensional space?