How can we derive the expression for cross product of two vectors?

Here is a video that shows where the term cross comes from.

Two vectors in a 3D space when “multiplied” by each other result in a third vector that represents how they “cross” each other.

The cross product accumulates interactions between different dimensions. Taking two vectors, we can write every combination of components in a grid:

https://betterexplained.com/wp-content/uploads/crossproduct/cross-product-grid.png
This completed grid is the outer product, which can be separated into the:

Dot product, the interactions between similar dimensions (x*x y*y, z*z)

Cross product, the interactions between different dimensions (x*y,y*z, z*x, etc.)

The dot product (a⃗ ⋅b⃗ a→⋅b→) measures similarity because it only accumulates interactions in matching dimensions. It’s a simple calculation with 3 components.

The cross product (written a⃗ ×b⃗ a→×b→) has to measure a half-dozen “cross interactions”. The calculation looks complex but the concept is simple: accumulate 6 individual differences for the total.

https://betterexplained.com/articles/cross-product/