Existence of orthogonal Hamel basis for infinite dimensional vector space.

Question

Consider a $\mathbb{F}$-vector space $V$ that is infinite dimensional and equip it with an inner product (i.e. it is a Pre-Hilbert space). We know that $V$ has a basis set, say $S$, such that for any $v\in V$, $v=\sum_{v_s\in S}a_sv_s$ where only finitely many $a_s\in\mathbb{F}$ are non-zero. For a finite dimensional vector space we can orthogonalize the basis vectors using (Gram-Schmidt) but is it possible to do the same in an infinite dimensional vector space? I feel that if the basis were countable, we could do this but when the basis is uncountable (like the free vector space generated by an uncountable set $S$) it might not be. Is it possible to find an orthogonal uncountably infinite basis?