Given the following term: $$\sum_{i<j} A_{i,j} x_i A_{j,i} x_j$$
(with $A$ a square matrix and $x$ a vector) is there a nice explicit matrix expression?
I've tried: $$\sum_{i<j} A_{i,j} x_i A_{j,i} x_j = \frac{1}{2}\sum_{i,j} A_{i,j} x_i A_{j,i} x_j - \sum_{i} A_{i,i}^2 x_i^2$$
as well as the identity $\sum_{i,j}(A B)_{i,j} = tr(A B)$.
Although it looks simple, I couldn't find an expression that doesn't involve a $\sum$.