Consider the category of chain complexes over a ring $R$.

We can show that $\text{Ext}^1(M, N)$ classifies extensions using the triangulated category structure: the homotopy kernel of a map $N \rightarrow M[1]$, $K$, gives rise to the designated triangle $M \rightarrow K \rightarrow N$. This gives a simple picture of Yoneda's extension theorem for $\text{Ext}^1(M, N)$, which also seems to rely on the "threeness" of a triangulated category.

My question is, is there an extension of this view to $\text{Ext}^n(M, N)$?

The proof of the situation for $\text{Ext}^1$ is quite simple using triangulated categories and it would be nice if the situation could be handled the same way for $\text{Ext}^n$.