You can use the general local formula for the Laplace-Beltrami operator in terms of any local orthonormal frame:

$$\Delta = \sum_{i=1}^n X_i^2 +\mathrm{div}(X_i)X_i$$

where the $X_i$'s are seen as derivations on functions.

You can always find a local frame of vector fields $X_1,\ldots,X_n$ that are divergence-free at a given point $q$. In terms of this frame, the Laplacian at the point $q$ is just a "sum of squares".

Locally, the construction of a local divergence-free, orthonormal frame leads to a system of first order PDEs. The integrability conditions then gives a local obstruction.

EDITED AFTER COMMENT, THANKS to Jean Van Schaftingen for pointing out an imprecision in my previous answer.