I'd like to know whether I'm thinking about this in the right way.
Let's say that I have a subspace of R^4 and have found an orthonormal basis for it that consists of three vectors. If I want to then find a basis for all of R^4 i need to find a vector that is not in the same linear span as the original three vectors and perform Gram-Schmidt. This is the same as finding a vector that is not a linear combination of the original three vectors. So, I would put all four vectors into a matrix (as rows) and then put it into rref. If the matrix has no all-zero rows, the four vectors are linearly independent, and therefore the vector I choose is not in the linear span of the first three vectors and can be used to find a basis for all of R^4