Sorry if this is a silly question. But if I have a polynomial in two variables, with the maximum degree of each individual variable no greater than N, $$p(x, y)=c_{00}+c_{10}x+c_{01}y+c_{11}xy+c_{20}x^2+c_{02}y^2+... =\sum_{i=0}^N{\sum_{j=0}^N{c_{ij}x^iy^j}}$$
...then I can write this as a product,
$$p(x,y)=\textbf{x}'A\textbf{y},$$
where
$$\textbf{x}=\begin{matrix}\begin{pmatrix}1&x&x^2&...&x^N\end{pmatrix}\end{matrix}'$$ $$\textbf{y}=\begin{matrix}\begin{pmatrix}1&y&y^2&...&y^N\end{pmatrix}\end{matrix}'$$ $$A=\begin{matrix}\begin{pmatrix}c_{00}&c_{01}&c_{02}&...&c_{0N} \\c_{10}&c_{11}&c_{12}&...&c_{1N} \\c_{20}&c_{21}&c_{22}&...&c_{2N} \\... \\c_{N0}&c_{N1}&c_{N2}&...&c_{NN}\end{pmatrix}\end{matrix}$$
This seems convenient, since the elements of A correspond exactly to the coefficients on each term of the function.
My questions are:
1) Is this a "standard" way of representing a polynomial of two variables?
2) Is there some convenient way to extend this to polynomials of more than two variables?
I don't know if there's a more general way to state this question, or if there is some theory out there regarding multivariate polynomials that I could learn, but any answers or links to resources would be helpful...I'm not quite sure what to Google for this.