## Number of Independent Elements in an Antisymmetric Matrix

The entries in am antisymmetric matrix satisfy
$a_{ij} = -a_{ji}$
.
This means that any entry on the main diagonal must be zero since it satisfies
$a_{ii} = -a_{ii}$
.
Any 3 x 3 antisymmetric matrix takes the form
$\left( \begin{array}{ccc} 0 & a_{12} & a_{13} \\ -a_{12} & 0 & a_{23} \\ -a_{13} & -a_{23} & 0 \end{array} \right)$
.
The independent entries are
$a_{12} \: a_{13}, \: a_{23}$
.
There are 3 independent elements. We cab take the three elements in the upper right and the three elements on the leading diagonal to be the independent elements.
In general for an
$n x n$
square matrix we can take the
$\frac{n(n-1)}{2}$
elements in the upper right to be the independent elements, so there are
$\frac{n(n-1)}{2}$
independent elements.