Is there a general rule for how to write high order polynomials in matrix form? 
Is there a general rule for how to write high order polynomials in matrix form? 

For example a linear combination of parameters:
$$a_1 x_1+a_2 x_2+a_3 x_3 + \cdots+ a_n x_n$$
Can be written as
$$\sum^n_{i=1} a_i x_i = \vec{a}^T\vec{x} $$
Second order forms are given by 
$$ (a_1 x_1+a_2 x_2+a_3 x_3 + \cdots+ a_n x_n)^2 = \vec{x}^T {\mathbf A} \vec{x}$$
Which ensures all combinations of second order terms. What about the higher orders? i.e.
$$(a_1 x_1+a_2 x_2+a_3 x_3 + \dots +a_n x_n)^k$$
What forms ensure all combinations of terms. Is there a general rule to this? Does it have a name?
 A: Linear Form

$$
\begin{pmatrix}
  a & b \\
\end{pmatrix}
\begin{pmatrix} x \\ y \end{pmatrix}=
ax+by$$

Quadratic Form

$$
\begin{pmatrix} x & y \end{pmatrix}
\begin{pmatrix}
  a & b \\
  b & c
\end{pmatrix}
\begin{pmatrix} x \\ y \end{pmatrix}=
ax^2+2bxy+cy^2$$

Cubic Form

$$
\begin{pmatrix} x & y \end{pmatrix}
\begin{pmatrix}
  ax+by & bx+cy \\
  bx+cy & cx+dy
\end{pmatrix}
\begin{pmatrix} x \\ y \end{pmatrix}=
ax^3+3bx^2y+3cxy^2+dy^3$$

Quartic Form

$$
\begin{pmatrix} x & y \end{pmatrix}
\begin{pmatrix}
  ax^2+2bxy+cy^2 & bx^2+2cxy+dy^2 \\
  bx^2+2cxy+dy^2 & cx^2+2dxy+ey^2
\end{pmatrix}
\begin{pmatrix} x \\ y \end{pmatrix}$$

$m$-tuple Form

$$ \binom{m}{i_{1},i_{2}, \ldots , i_{n}} a_{i_{1} i_{2} \ldots i_{n}} x_{1}^{i_{1}} x_{2}^{i_{2}} \ldots x_{n}^{i_{n}}$$
where $\boldsymbol{x} \in \mathbb{R}^{n}$ and $i_{1}+i_{2}+\ldots+i_{n}=m$

A: I like what has been said already by @Daniel and @Ng Chung Tak but let me rephrase their answers.
We'll consider tensor products of the 1-form $a=(a_1,\ldots a_n)\in V^*$. A bilinear form $b=a\otimes a$ (which we'll call a "second order form" when supplied with the same $x\in V$ in both argument slots) can indeed be represented by a matrix equal to the outer product of the coordinates of $a$. A multilinear form $m=\otimes^ka$ would be represented by a multidimensional array.
Alternatively, you can look at this as an application of Multinomial theorem and multi-index notation
$$
(a_1x_1+\cdots+a_nx_n)^k = \sum_{|\alpha|=k}{k \choose \alpha}a^\alpha x^\alpha
$$
