I want to know if it is possible to express the operation

$$ \nabla \phi \times (\nabla \times \mathbf A) $$

as the divergence of second order tensor field $T$. Here $ \phi$ is a scalar field and $\mathbf A$ is a solenoidal vector field ($\nabla \cdot \mathbf A=0$)

I have used all possible identities and finally I can only get

$$ \nabla \phi \times (\nabla \times \mathbf A) = \nabla \phi \cdot(\nabla \mathbf A - \nabla \mathbf A^T) $$

Is it possible to find something like $ \nabla \phi \times (\nabla \times \mathbf A) = \nabla \cdot(\text{tensor}) $ ?