Chain rule for the curl of a vector-valued function

Question

I am looking for a vector expression for the curl of a composite vector-valued function. In other words $$ \nabla\times\mathbf{A}(\mathbf{B}) = ? $$ In indicial notation, this can be written as $$ \left(\nabla\times\mathbf{A}(\mathbf{B})\right)_i = -\epsilon_{ijk}\left(\frac{\partial A_k}{\partial B_n}\right)\frac{\partial B_n}{\partial x_j} $$ where $\epsilon_{ijk}$ is the Levi-Civita symbol and Einstein summation is used. so far, the simplest expression we could write was
$$ \left(\nabla\times\mathbf{A}(\mathbf{B})\right)_i = \epsilon_{ijk}\left[\left(\frac{\partial\mathbf{A}}{\partial\mathbf{B}}\right)\cdot\left(\nabla\otimes\mathbf{B}\right)^T\right]_{jk} $$ where $\cdot$ is used to denote the usual matrix product and $\otimes$ the Kronecker outer product. Could this be written all in vector notation?