I am very far from an expert on the subject, but I think the Hochschild homology of the coordinate ring should be the algebraic de Rham complex of your variety SO(n, R)--not the cohomology of the complex, just the groups in the complex (with 0 differential if you like). This is the Hochschild-Kostant-Rosenberg theorem and should just require smoothness, not the fact that you have a Lie group.
I don't know what HH^{alt} is, but maybe you can figure it out from this description. Unless I am very confused about something, HH_k is nonzero exactly for 0 <= k <= the dimension of SO(n, R), so your guess about d sounds plausible.
