Q:

Product of a symmetric and anti-symmetric matrix

I have the following question about matrices,
Let $S$ and $A$ be two $n \times n$ matrices which are respectively symmetric and anti-symmetric. Can I conclude anything about the products $SA$ or $AS$, are they symmetric or anti-symmetric? 
This is part of a bigger problem where I have already shown,
$$ \langle x, Ax \rangle = 0$$
For $A$ antisymmetric, but I require that 
$$ \langle Mx, Ax \rangle = 0$$
For some matrix M, what conditions could I impose on $M$ to satisfy this, I was hoping symmetry would be sufficient or do I require something stronger such as diagonality? 

A:

The matrix product does not preserve the symmetric nor the anti-symmetric property. A simple example of this phenomenon is the following.
Pick
$$S=\begin{pmatrix}
2 & 1\\
1 &2
\end{pmatrix} \quad \text{and} \quad A=
\begin{pmatrix}
0 & -1\\
1 &0
\end{pmatrix}.$$
Then $$SA=\begin{pmatrix}
2 & 1\\
1 &2
\end{pmatrix}
\begin{pmatrix}
0 & -1\\
1 &0
\end{pmatrix}=
\begin{pmatrix}
1 & -2\\
2 &-1
\end{pmatrix}$$
which is symmetric nor anti-symmetric. Similarly,
$$AS=
\begin{pmatrix}
0 & -1\\
1 &0
\end{pmatrix}\begin{pmatrix}
2 & 1\\
1 &2
\end{pmatrix}=
\begin{pmatrix}
-1 & -2\\
2 &1
\end{pmatrix}$$
Again, this is not symmetric nor anti-symmetric.

