On discrete values of bilinear forms

Let $\omega$ be a nondegenerate skew-symmetric bilinear form in the real plane. We prove that for finite a point set $P\subset \mathbb R^2\setminus\{0\}$, the set $T_\omega(P)$ of nonzero values of $\omega$ on $P\times P$, if nonempty, has cardinality $\Omega(N^{96/137})$.
In the special case when $P=A\times A$, where $A$ is a set of at least two reals, we establish the following sum-product type estimates, corresponding to the symmetric and skew-symmetric form $\omega$:
$$|AA+ AA|= \Omega(|A|^{19/12}) \quad\text{and}\quad |AA-AA|= \Omega\biggl( \frac{|A|^{49/32}}{\log^{3/32}|A|}\biggr).$$
These estimates improve their basic prototypes $\Omega(N^{2/3})$ and $\Omega(|A|^{3/2})$, which readily follow from the Szemerédi-Trotter theorem.
Bibliography: 28 titles.