I eventually solved it, though I'm not sure it's the shortest path.
Perpendicularity and orthogonality is the same. You can say that two vector spaces V,W are perpendicular if all vectors v1,..,vn in V are perpendicular to all vectors w1,..,wm in W. Which is defined over the orthogonality derived from the dot product.
In that manner, the kernel of a matrix A is orthogonal to the row space of A. To find the basis of the kernel, I need to get the transpose of A into rref, thus finding the basis for the row space and from that I can also read the basis of the kernel.
With the regular rref I could only find out the basis of the "left kernel" (aka the kernel of the column space).
Found the answers in lecture 9 and 10 of MIT 18.06: https://www.youtube.com/watch?v=ZK3O402wf1c&list=PLE7DDD91010BC51F8