How do I prove that the standard basis in $\mathbb{F}^n$ ($\mathbb{R}^n$ or $\mathbb{C}^n$) is orthonormal?

Question

I have seen that people just use the Euclidean inner product on $\mathbb{F}^n$. In that case I can see clear that the norm of every vector is $1$ and are pairwise orthogonal. But I wonder if I just can assume that specific inner product on $\mathbb{F}^n$ or it can be proven on every general inner product on the vector space just using the definition of inner product on a general vector space. How do I prove it in general?