I still don’t know the map between a Clifford algebra Cn and rotations of Rn, say as orthogonal matrices. They both compose thru their respective multiplications, but this fact doesn’t yield a direct map. I know the map between C3 and rotations of R3. Only C values comprised of even terms represent rotations. That subspace of C3 is isomorphic to the quaternions. It uses only the unit quaternions but the others can be employed as well to represent expansions and contractions along with rotations (See conformal matrix here).

Perhaps I could factor orthogonal matrices into products for each of which I knew the Clifford number. Perhaps I could do the opposite but that seems harder, especially since Clifford algebras are not division algebras.

Factoring an orthogonal matrix into simple rotations is not unique. Would different factorizations produce different Clifford numbers, beyond the sign ambiguity? That sounds like a feasible programming project!

Given an orthogonal matrix (OM) we multiply it, on left and right, by a sequence of standard OMs to put it into a form where it is a matrix of zeros except for 2 by 2 matrices down the diagonal of the form:

We use the following standard matrices:
(Mmn(θ))ij =
if i=j=m or i=j=n thencosθ
if i=m and j=n then sinθ
if i=n and j=m then −sinθ
otherwise if i=j then 1
I think I know the Clifford numbers for such rotations.