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?

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 $i_{1}+i_{2}+\ldots+i_{n}=m$

