I am currently reviewing some algebra and I recalled that cross product can be defined in terms of a determinant: i,j and k are described as being an orthonormal basis. We can take i, j and k as the following: i = {1,0,0} j = {0,1,0} k = {0,0,1} However each of these are 3-d vectors, how can they "fit" into their respective matrix positions while preserving the shape of the matrix?

The operation requires multiplication of vectors by scalars, and addition of vectors, which are both valid operations. That form is just a short hand to help remember the cross product formula. "Fitting" has nothing to do with it.

It's not a real determinant, just an interesting shorthand / memory aid / coincidence. Just pretend that i, j, and k are normal scalar variables.