A property of a symmetric 2 elements field matrix with zero diagonal

Question

Let $A$ be a symmetric $m \times m$ matrix over the two element field all of whose diagonal entries are zero. Then the bilinear form $(w, Aw) = {w^T}Aw = 0$ for all vectors $w$ over the same field.

I'm really confused with this statement, anyone have idea to prove it?

This is a sub-statement of problem 4 of 2010 International Mathematics Competition for University Students. Just make sure someone want to know it's resource.

Answer 1

$$ (w,Aw) = \sum_{i,j}w_iA_{i,j}w_j = $$$$= 2\sum_{i<j}w_iA_{i,j}w_j=0. $$