Deriving the bivariate polynomial basis functions

Question

Is there a means of computing the bivariate polynomial basis functions for an $n^{th}$-order polynomial from the associated $n^{th}$-order univariate polynomial basis functions? E.g. How is $\{1, x, y, x^2, xy, y^2\}$ computed from $\{1, x, x^2\}$ and $\{1, y, y^2\}$. I have done a bit of searching, but have not found a definite answer to this. My initial guess was that it would involve the outer product, but this produces an $x^2y^2$ term.