Gram-Schmidt in characteristic two?

Question

I was helping someone work on a computing problem with bit vectors that reduced to finding a basis knowing a spanning set, and realized quickly that the Gram-Schmidt process does not work as expected in characteristic-2. Since orthogonality was not a requirement anyway I came up with an alternate approach, but it left me wondering: Is there a way to salvage Gram-Schmidt in this setting (a vector space over $\mathbb{F}_2 = \mathbb{Z}/2\mathbb{Z}$) to get a meaningful concept of orthogonal basis? (Orthonormal seems out of the question since I don't see a way to define a meaningful norm.)