Metric for how symmetric a matrix is
Solution 1:
One simple possibity:
$s \equiv (|A_{sym}|-|A_{anti}|)/(|A_{sym}|+|A_{anti}|)$
Here |·| is whatever matrix norm you choose. Then $-1\le s \ \le +1$ with the lower bound saturated for an antisymmetric matrox, upper bound saturated for a symmetric one.