If M is skew symmetry $N^{T}MN$  is skew symmetry

     If M is symmetry $N^{T}MN$  is  symmetry

Option B) ${\left(MN-NM\right)}^T={\left(MN\right)}^T-{\left(NM\right)}^T=N^T{M}^T-M^T{N}^T$

                 If M and N are symmetry

                   ${\left(MN-NM\right)}^T=NM-MN=-\left(MN-NM\right)$

                 $MN-NM$  is skew symmetry

Option C) ${\left(MN\right)}^T=N^T{M}^T=NM$

                  MN is not symmetry for symmetric matrices M and N

Option D) (adj M)(adj N)=adj(NM)

                   $\therefore \left(\text{adj M}\right)\left(\text{adj N}\right)\ne \text{adj}\left(\text{MN}\right)$