Since, $\mathrm{A}$ is a skew symmetric matrix
$\therefore {\mathrm{A}}^{\mathrm{T}}=-\mathrm{A}$
$\Rightarrow \left|{\mathrm{A}}^{\mathrm{T}}\right|=|-\mathrm{A}|$
$\Rightarrow |-\mathrm{A}|={\left(-1\right)}^{\mathrm{n}}|\mathrm{A}|$
where $\mathrm{n}$ is the order.
$\therefore |-\mathrm{A}|={\left(-1\right)}^3|\mathrm{A}|$
$=-|\mathrm{A}|$
Substituting in (i),
$\left|{\mathrm{A}}^{\mathrm{T}}\right|=-|\mathrm{A}|$
$\Rightarrow -|\mathrm{A}|=\left|{\mathrm{A}}^{\mathrm{T}}\right|$
We also know for any matrix.
$\left|\mathrm{A}\right|=\left|{\mathrm{A}}^{\mathrm{T}}\right|$
replacing in (ii)
$\therefore |\mathrm{A}|=-|\mathrm{A}|$
$\Rightarrow 2|\mathrm{A}|=0$
$\Rightarrow |\mathrm{A}|=0$