Orthogonal Basis for Partial Differentiation?

Question

Does taking partial dervatives in $\mathbb{R}^n$ acutally requrire us to take an orthogonal basis? I know the partial derivative as directional derivative in the direction of a basis vector, but nothing about that seems to use a dot product. A bunch of authors seem to require it, but is there something that goes wrong (other than we can't draw pretty pictures anymore) if we take an arbitrary basis?