Any orthonormal set extends to an orthonormal basis, over any field of characteristic not $2$.
Proof: Let $q_n$ be the quadratic form $x_1^2+\cdots+x_n^2$. Translated into the language of quadratic forms, your question is: Given a quadratic form $q$ such that the orthogonal sum $q \perp q_m$ is equivalent to $q_n$ for some $m \le n$, must $q$ be equivalent to $q_{n-m}$? The answer is yes, by the Witt cancellation theorem.