I was reading this post: Deriving formula for cross-product. and I understand the derivation there of the formula and it's easy enough to just manually check that the resulting formula is the same as the one we get from taking the determinant of the matrix with the first row being the unit vectors in $\mathbb{R}^3$ and the second two rows being the Cartesian coordinates of the two vectors we're crossing.
But is there a way to derive that determinant expression more directly, perhaps by thinking in terms of linear transformations instead of by solving a system of equations via elimination?
To be clear, I'm aware it's often said that writing the formula as a determinant is an abuse of notation meant only as a mnemonic, and that we aren't really taking a determinant. It's unclear to me why that would be the case though. I mean, I get that the elements of the top row of the matrix are each members of $\mathbb{R}^3$ while the elements of the second two rows are members of $\mathbb{R}$, which presumably means the matrix is a representation of a linear transformation not on $\mathbb{R}^3$ but on some vector space that's a combination of $\mathbb{R}^3$ and $\mathbb{R}$, right? But there surely must still be some way to rigorously and directly derive this expression.
Edit: I mean "derive" in the sense of a direct proof (i.e. not a proof by contradiction or one that relies on manually verifying that two expressions happen to be equal).