\begin{split}
 a \wedge b
 &= ( a_1 e_1 + a_2 e_2 )
 \wedge ( b_1 e_1 + b_2 e_2 ) \\
 &= \begin{vmatrix}
 a_1 & b_1 \\
 a_2 & b_2
 \end{vmatrix} ( e_1 \wedge e_2 )
\end{split}

\begin{split}
 a \wedge b
 &= ( a_1 e_1 + a_2 e_2 + a_3 e_3 )
 \wedge ( b_1 e_1 + b_2 e_2 + b_3 e_3 ) \\
 &= \begin{vmatrix}
 a_1 & b_1 \\
 a_2 & b_2
 \end{vmatrix} ( e_1 \wedge e_2 )
 + \begin{vmatrix}
 a_2 & b_2 \\
 a_3 & b_3
 \end{vmatrix} ( e_2 \wedge e_3 )
 + \begin{vmatrix}
 a_3 & b_3 \\
 a_1 & b_1
 \end{vmatrix} ( e_3 \wedge e_1 ) \\
 &= \begin{vmatrix}
 a_1 & b_1 & e_{23} \\
 a_2 & b_2 & e_{31} \\
 a_3 & b_3 & e_{12}
 \end{vmatrix}
\end{split}