Q:

Find if a form is symmetric or skew-symmetric

Consider the set of all n × n matrices in R. Given the defined function 
Φ: $M$(n,n)× $M$(n,n) → R , which
Φ(A,B) = $tr$(A$^T$JB) ,
where J is a skew-symmetric n × n matrix , define if Φ is a bilinear form, and in case it is define if it is symmetric or skew-symmetric.
So far, I've already proved that Φ is a bilinear form. I know that a form is symmetric if: 
Φ(A,B) = Φ(B,A)
and skew-symmetric if
Φ(A,B) = -Φ(B,A)
I tried to prove it by doing:
Φ(A,B) = $tr$(A$^T$JB)
Φ(B,A) = $tr$(B$^T$JA)
but I'm not sure how to proceed in order to prove if Φ is symmetric or skew-symmetric.

A:

Hint:
$$
\DeclareMathOperator{\tr}{tr}
\tr(A^TJB) = \tr([A^TJB]^T) = 
\tr(B^TJ^TA) = -\tr(B^TJA)
$$

