Connection between cross product and determinant

Maybe this isn't the answer you're looking for, but one expression for the determinant of a 3x3 matrix with columns $\vec v_1,\vec v_2,\vec v_3$ is $$ \vec v_1\cdot(\vec v_2\times\vec v_3) $$ You can make sense of this algebraically or geometrically (recall that the determinant is the volume of a parallelipiped whose sides are given by the three vectors).

One definition of the cross product is the vector $a \times b$ such that $\langle x , a \times b \rangle = \det \begin{bmatrix} x & a & b\end{bmatrix}= \det \begin{bmatrix} x^T \\ a^T \\ b^T\end{bmatrix}$.

This is, of course, equivalent to all of the above.

To determine the $x,y,z$ components of $a \times b$ one computes $\langle e_k , a \times b \rangle$ for $k=1,2,3$ which gives, of course, exactly the same answer as the symbolic version with $x^T = ( i, j , k )^T$.

If $\vec{i},\vec{j},\vec{k}$ are the three basic vectors of $\mathbb{R}^3$ then the cross product of vectors $(a,b,c), (p,q,r)$ is the determinant of the matrix $$\left(\begin{array}{lll}\vec{i}&\vec{j}&\vec{k}\\ a &b & c\\ p&q &r\end{array}\right)$$ by definition. The coordinates of that vector are obtained by expanding this determinant along the first row.