Let's work in the category $R$-${\sf mod}$, for concreteness. I know that one can see the modules ${\rm Ext}^n_R(M,N)$ as modules of equivalence classes of $n$-extensions of $M$ by $N$ (Yoneda extensions), namely, exact sequences of the form $$0 \to N \to E_1 \to \cdots \to E_n \to M \to 0,$$with certain operations (more precisely, if one denotes such collection by ${\rm E}^n(M,N)$, there are natural isomorphisms ${\rm E}^n(M,N)\cong {\rm Ext}_R^n(M,N)$ for each $n$).

Is there anything similar for ${\rm Tor}^R_n(M,N)$?

I expect the answer to be highly non-trivial, for the following analogy: this business about $n$-extensions effectively gives us a way to describe the elements of ${\rm Ext}^n_R(M,N)$, but why should we expect any simple explanation for ${\rm Tor}$ if we cannot even describe the elements of ${\rm Tor}_0^R(M,N) = M\otimes_RN$ in general?

Apologies if by any chance this is a repeated question, a quick search on the website didn't show up anything here.