homological algebra - About higher Ext in R-Mod

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 $0in Hom_R(A,B)congmathrm{Ext}^0(A,B)$, 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 $Arightarrow Aoplus Brightarrow B$, but precisely what sort of extension? Or is that already completely wrong?

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