Twisted tensor products of $K^n$ with $K^m$

We find three families of twisting maps of K^m with K^n. One of them is related to truncated quiver algebras, the second one consists of deformations of the first and the third one requires m=n and yields algebras isomorphic to M_n(K). Using these families and some exceptional cases we construct all twisting maps of K^3 with K^3.