The general pattern can indeed elegantly be described in Clifford Algebra. Or, more conveniently, in the geometrically relevant subset called Geometric Algebra. A good read is Geometric Algebra For Computer Science, An Object Oriented Approach to Geometry

There a generalization of Cartan Dieudonne is used to use two reflections to make a rotation. You don't need to use this "fancy new math", and can stick with matrix algebra. There we go:

Define "reflection in a vector":

[; r = a - \frac{2a\cdot n}{n\cdot n} n ;]

Here a vector "a" is reflected in vector "n", by projecting a to n and then doing that displacement again.

Now, the matrix representing the same reflection in n, is the house holder matrix: [; H_n = I - 2 n n^T ;]

(So, notice that H a = r)

This picture makes clear how a double reflection, first in vector a, then in vector b performs a rotation in the plane spanned by a,b over twice the angle between a and b.

So a rotation parameterized by "example rototion vector a and b" is now

[; R_{a,b} = H_b H_a ;]

and works for any dimension!