Compute the index of the inner product space $\Ma(n,\R)$, $\la A,B\ra\colon=\tr(AB)$.
Decompose $\Ma(n,\R)=E\oplus F$, where $E$ is the subspace of all skew-symmetric matrices and $F$ the subspace of all symmetric matrices. For all $A\in E$ and all $S\in F$ we have
$$
\tr(AS)=\tr((AS)^t)=\tr(S^tA^t)=-\tr(SA)=-\tr(AS)~.
$$
Hence $\tr(AS)=0$. Analogously for all $A\in E\sm\{0\}$: $\tr(AA)=-\tr(AA^t) < 0$ and for all $S\in E\sm\{0\}$: $\tr(SS)=\tr(SS^t) > 0$. Therefore the index is $\dim(E)=\frac12n(n-1)$.