Objective to find visual and accessible ways to remember this formula fast
$(x,y,z)\times(u,v,w)=(yw-zv,zu-xw,xv-yu)$
I have used Sarrus' rule but it is slow, more here. Since it is slow, I have tried to find alternative ways such as binary-tree -visualization (but it is poor/slow until some clever ideas):
RobJohn's idea to "just cycle the permutation"
$\begin{align}i&=j\times k\\j&=k\times i\\k&=i\times j\end{align}$
Mariano Suárez‐Alvarez -idea
View the 3×3 matrix as the points of the affine plane over F3: then the terms in the determinant correspond to affine lines which are neither horizontal nor vertical --.
Anon's idea
Draw a triangle with vertices i, j, k with arrows i->j, j->k, k->i. Multiplying two of these units is done as follows: if they are the same, 0; if they go with the flow of the triangle, the third in the line; if the go opposite the flow, the third one as well but with a minus sign
We are discussing this issue more here.