If $T$ is a *linear* map on a real vector space, it is completely determined by the way it maps any basis of the vector space (by linearity).

This is the general gist of matrix notation - by specifying the image of the linear transformation on $(1,0,0,...), (0,1,0,...), (0,0,1,...), ...$, the linear transformation is uniquely determined.

Is there an analogous concept for (unital, associative) algebras? Something like a minimal set of vectors such that specifying an algebra homomorphism on this set determines it for all others?

I'd be happy enough to answer this just for the tensor algebra, the symmetric algebra, or the exterior algebra, since these would be generally sufficient for my purposes.