Is there is a test for determining if a rank 1 tensor is the divergence of a symmetric rank two tensor?

Question

Suppose that $A^i$ is a rank 1 tensor.

Is there a test for checking if \begin{equation} A^i = \partial_j\tau^{ji} \end{equation}

where $\tau^{ij} = \tau^{ji}$?

An analogous test would be checking if a vector is the gradient of some scalar by taking its curl.