I have been trying to construct a orthogonal basis of $\mathbb{F}_{2}^{n}$ for an odd value of $n$ which is comprised of vectors which are not $1$ (to avoid the standard basis). In particular, to mirror the assumption on $n$, I want to ensure that the basis has only vectors with an odd number of $1$'s.
Is this possible? I have tried to prove otherwise, however have run out of ideas.