If $x_1=\sum_{k=j}^n a_{kj}e_k$ , then for every alternating $n$-form $\o$ on the $n$-dimensional vector-space $E$: $\o(x_1,\ldots,x_n)=a_{11}\ldots a_{nn}\o(e_1,\ldots,e_n)$.
By the previous examples
\begin{eqnarray*}
\o(x_1,\ldots,x_n)
&=&\o(x_1,\ldots,a_{n-1,n-1}e_{n-1}+a_{n,n-1}e_n,a_{nn}e_n)\\
&=&\o(x_1,\ldots,a_{n-1,n-1}e_{n-1},a_{nn}e_n)\\
&=&\o(x_1,\ldots,a_{n-2,n-2}e_{n-2}+a_{n-1,n-2}e_{n-1}+a_{n,n-2}e_{n},a_{n-1,n-1}e_{n-1},a_{nn}e_n)\\
&=&\o(x_1,\ldots,a_{n-2,n-2}e_{n-2},a_{n-1,n-1}e_{n-1},a_{nn}e_n)
=\cdots=\o(a_{11}e_1,\ldots,a_{n-1,n-1}e_{n-1},a_{nn}e_n)
\end{eqnarray*}