= Solution
{c}
Let $A$ be a <skew-symmetric matrix>. For every real <vector> $x$, the real <scalar> $x^TAx$ equals its own <transpose>, but
$$
x^TAx=(x^TAx)^T=x^TA^Tx=-x^TAx.
$$
Hence $x^TAx=0$. If $Av=\lambda v$ is a real <eigenvector> equation with $v\ne0$, it follows that
$$
0=v^TAv=\lambda v^Tv,
$$
so $\lambda=0$. If $A$ is a <diagonalisable matrix> over $\mathbb R$, it has a real <basis> of <eigenvectors>, all with <eigenvalue> zero. Thus $A$ vanishes on a <basis> and $\boxed{A=0_n}$.
The <field> matters: a nonzero real <skew-symmetric matrix> can be a <diagonalisable matrix> over the <complex numbers>, for example $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ with <eigenvalues> $i,-i$.
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