Find the matrix that associated to a given quadratic form

Question

Please help me complete my answer and find the matrix associated to a given quadratic form, here is my work and effort on that

Here's my attempt to answer it

Answer 1

HINT

Recall that for a skew symmetric matrix $S$

$$v^TSv=0 \quad \forall x$$

and that

$$v^T(A+S)v= v^TAv+v^TSv$$