Let $I = \langle f_1, \ldots f_r \rangle$ be an ideal in $R=k[x_1,\ldots,x_n]$ where $k$ is a field, and put $A = R/I$.

(If $I$ is prime then $A$ is the coordinate ring of an irreducible affine variety.)

Let $\mathfrak{X}^p(A) = \operatorname{Der}_k(\wedge^p A, A) $ be skew-symmetric $p$-derivations (derivation in each argument) of $A$.

Question: Is there an algorithm to find (generators and relations for) $\mathfrak{X}^p(A)$ in terms of $f_1,\ldots,f_r$?

I asked this question on Math.SE and eventually ~~only partially~~ answered it myself.

The algorithm in my partial answer finds only the module consisting of those elements of $\mathfrak{X}^p(A)$ which lift to $\mathfrak{X}^p(R)$. Is this algorithm (or any alternative algorithm) known and/or written down in the literature somewhere? Is there any algorithm that works more generally?