Remember cofactor expansion:
$$\det \left[ {\begin{matrix}
{\color{red}0} & {\color{red}0} & \ldots & {\color{red}0} & {\color{red}1}\\
0 & 0 & \ldots & 1 & 0\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
0 & 1 & \ldots & 0 & 0\\
1 & 0 & \ldots & 0 & 0\\
\end{matrix} } \right] = $$
$${\color{red}0}\cdot \det \left[ {\begin{matrix}
0 & \ldots & 0 & 1\\
\vdots & \ddots & \vdots & \vdots\\
0 & \ldots & 0 & 0\\
1 & \ldots & 0 & 0\\
\end{matrix} } \right] -
{\color{red}0}\cdot \det \left[ {\begin{matrix}
0 & \ldots & 0 & 1\\
\vdots & \ddots & \vdots & \vdots\\
0 & \ldots & 0 & 0\\
1 & \ldots & 0 & 0\\
\end{matrix} } \right] +
{\color{red}0}\cdot \det \left[ {\begin{matrix}
0 & \ldots & 0 & 1\\
\vdots & \ddots & \vdots & \vdots\\
0 & \ldots & 0 & 0\\
1 & \ldots & 0 & 0\\
\end{matrix} } \right] +\cdots $$
$$\cdots + (-1)^{n-2}{\color{red}0}\cdot \det \left[ {\begin{matrix}
0 & \ldots & 0 & 0\\
\vdots & \ddots & \vdots & \vdots\\
0 & \ldots & 0 & 0\\
1 & \ldots & 0 & 0\\
\end{matrix} } \right]
+ (-1)^{n-1}{\color{red}1}\cdot \det \left[ {\begin{matrix}
0 & \ldots & 0 & 1\\
\vdots & \ddots & \vdots & \vdots\\
0 & \ldots & 0 & 0\\
1 & \ldots & 0 & 0\\
\end{matrix} } \right] $$
So the only thing that survives is
$$(-1)^{n-1}\cdot \det \left[ {\begin{matrix}
0 & \ldots & 0 & 1\\
\vdots & \ddots & \vdots & \vdots\\
0 & \ldots & 0 & 0\\
1 & \ldots & 0 & 0\\
\end{matrix} } \right] $$
Performing this $n$ times you will get $(-1)^{n(n-1)/2}$