In general coordinates, the basis vectors are defined as the derivatives of teh posituon vector: $$\vec e_i = \frac { {\partial \vec R}} {\partial x^i}$$
Nonetheless, the position vector is given by:
$$\vec R= \sum_i x^i \vec e_i$$
and when we do the derivative we have:
$$\frac{\partial \vec R}{\partial x^n} = \sum_i(\frac{\partial x^i}{\partial x^n} \vec e_i + x^i \frac{\partial \vec e_i}{\partial x^n}) = \vec e_n + \sum_ix^i\frac{\partial \vec e_i}{\partial x^n} $$
Since the term $\sum_ix^i\frac{\partial \vec e_i}{\partial x^n}$ is not null in general, then the question is: where am I messing up?
You should be considering $\vec R=\sum_i R^i\vec e_i$ (instead of $R=\sum_i x^i\vec e_i$) where each $R^i$ are functions of the euclidean coordinates $x^j$. It is necessary that the Jacobian matrix $J\vec R=\left[\frac{\partial R^i}{\partial x^j}\right]$ has determinant different from zero in the domain of definition of $R$.
A new basis for the space in these coordinates are $$\vec \partial_k=\sum_s\frac{\partial R^s}{\partial x^k}\vec e_s,$$ which give you, may be, a non-orthogonal neither non-orthonormal frame of the space.