So, in $R-Mod$, we have the rather short sequence

$\mathrm{Ext}^0(A,B)\cong Hom_R(A,B) $

$\mathrm{Ext}^1(A,B)\cong \mathrm{ShortExact}(A,B)\mod \equiv $, equivalence classes of "good" factorizations of $0\in Hom_R(A,B)\cong\mathrm{Ext}^0(A,B)$, with the Baer sum.

Question:

- $\mathrm{Ext}^{2+n}(A,B) \cong\ ??? $

While I suppose one could pose a conjugate question in algebraic topology/geometry, where the answer might look "simpler", I'm asking for a more directly algebraic/diagramatic understanding of the higher $\mathrm{Ext}$ functors. For instance, I'd expect $\mathrm{Ext}^2(A,B)$ to involve diagrams extending the split exact sequence $A\rightarrow A\oplus B\rightarrow B$, but precisely *what sort* of extension? Or is that already completely wrong?