Prove that every square matrix can be uniquely expressed as a sum of a symmetric matrix and a skew symmetric matrix.

$A+A^1$ is symmetric and  $A-A^1$ is a skew symmetric matrix and

$A+\frac{1}{2}\left(A+A^1\right)+\frac{1}{2}\left(A-A^1\right)$

To prove uniqueness let $B$ be a symmetric matrix and $C$ be a skew symmetric matrix such

$A=B+C$

Then $A^1=(B+C{)}^1=B^1+C^1=B-C$

and hence $B=\frac{1}{2}\left(A+A^1\right)$  and  $C=\frac{1}{2}\left(A-A^1\right)$