Let $\mathcal{P}$ be a (small) exact category. Without delving into any homotopy theory, we can provide characterisations of $K_0(\mathcal{P})$ and $K_1(\mathcal{P})$ as plain categorical constructions:

$K_0(\mathcal{P})$ is the free abelian group on the objects of $\mathcal{P}$ under the relations $[P] = [P'] + [P'']$ if $0 \to P' \to P \to P'' \to 0$

$K_1(\mathcal{P})$ is the free abelian group on the pairs $(P, \alpha)$, where $P$ is an object and $\alpha$ one of its automotphisms, under suitable relations.

Does $K_2(\mathcal{P})$ admit a similar description?

My question has been answered (and then some!). It has been pointed out however that I was slightly mistaken: the $K_1(\mathcal{P})$ given above only coincides with the standard one ($\pi_2(Q \mathcal{P})$) when short exact sequence of $\mathcal{P}$ splits. So I actually refer to the `Bass $K_1$'.