Let us suppose we have a, say, 10 dimensional real space with 3 orthogonal unit vectors given. How do I complete this orthonormal system with 7 additional vectors into a complete ONS in a way that is numerically stable?

An approach I could think of is to take a random vector, then perform a step in the Gram-Schmidt method to obtain a random unit vector that is orthogonal to all the previous ones. But I am concerned that choosing such a vector arbitrarily can introduce numerical errors which continue growing until I add the last vector to the system.

Is there a way I can choose the additional vectors in a way that I minimize numerical error?