A vector calculus formula

Question

Let $A, B$ be vector fields in $\mathbb R^3$. We have $$ \text{curl}\bigl((A\cdot \nabla)B\bigr)=(A\cdot \nabla)\text{curl}B -((\text{curl}A)\cdot \nabla)B+R(A,B). $$ I know that $R(A,A)=0$ and I guess that $R$ should resemble to $$ d a_j\wedge d b_j =\text{curl} A \ \times \text{curl} B+\text{div} A\ \text{curl} B -\text{div} B\ \text{curl} A.$$ Is there a simple closed form for $R(A,B)$?