I think I understand roughly what you mean when you distinguish row and column vectors. For example, let [x, y] represent a row vector and [x, y]^T represent a column vector.
If you have a function f that maps [x, y]^T -> z, (you might write it z=f(x,y)), then the gradient(f) is a function [x, y]^T -> [x, y]. That is to say, the gradient is a row vector. It's a different kind of vector than the input to f. And it transforms different (c.f. https://math.stackexchange.com/a/3200912/)
As you say, Geometric Algebra doesn't talk about row vectors and column vectors. For example, in 3D GA,you can choose a representation in R^8. That's 1 scalar, 1 pseudo-scalar, 3 column-y components, and 3 row-y components.
lol, ok - so then it's more the opposite - in GA there are row an column "vectors" (3-components)... but they are not "enough" (for that R^8 representation), so a matrix representation might be misleading ?
So using that kind of thinking, the cross product of two column vectors is a row vector: https://www.youtube.com/watch?v=BaM7OCEm3G0
As you say, Geometric Algebra doesn't talk about row vectors and column vectors. For example, in 3D GA,you can choose a representation in R^8. That's 1 scalar, 1 pseudo-scalar, 3 column-y components, and 3 row-y components.