On Wikipedia it is stated that any $n\times n$ real symmetric matrix A determines a quadratic form. But isn't $ax^2 + bxy + cxy + dy^2$, the quadratic form given by $v^T A v$ with $A=\begin{bmatrix}a & b \\ c & d\end{bmatrix}$ and $v=\begin{bmatrix} x \\ y\end{bmatrix}$, a quadratic form even when $c \neq b$?
Why does the Wikipedia article state that the matrix has to be symmetric?
It simply means that the matrix $A$ can be made symmetric without loss of generality. Simply define the off-diagonal elements of $A$ in your example to be $(b+c)/2$.