This is probably something very simple but I got stucked with it. Consider the polynomial ring $k[x_1,\dots,x_n]$. How can one construct a free resolution of it in the category of $k$-algebras(not necessarily commutative)? Namely, I ask for an acyclic complex $$\dots\to R_2\to R_1\to k[x_1,\dots,x_n]=R_0\to 0$$ where all $R_i$ are free algebras of some number of variables.
It is clear how one should try to construct such resolution: take $R_1=k\langle x_1,\dots, x_n\rangle$ with evident map to $R_0$. The kernel of this map is clearly generated by $x_ix_j-x_jx_i$ so we take $R_2=k\langle x_{ij}\rangle$ with differential taking $x_{ij}$ to $x_ix_j-x_jx_i$. And here is a problem I can not resolve - how to determine the kernel of this map and proceed futrther?