1. Use the fact that skew-symmetric matrices satisfy $M^T=-M,\qquad N^T=-N.$

2. Interpret the given expression

   The expression is $M^2N^2(M^TN)-^{-1}(MN^{-1})^T.$

   From the notation in the question, this is naturally read as $M^2N^2(M^TN)^{-1}(MN^{-1})^T.$

3. Simplify $M^TN$

   Since $M^T=-M$, $M^TN=(-M)N=-MN.$ Therefore, $(M^TN)^{-1}=(-MN)^{-1}=-(MN)^{-1}.$

4. Simplify $(MN^{-1})^T$

   Using $(AB)^T=B^TA^T$, $(MN^{-1})^T=(N^{-1})^TM^T.$

   Now, $N^T=-N \implies (N^{-1})^T=(N^T)^{-1}=(-N)^{-1}=-N^{-1},$ and also $M^T=-M$. Hence, $(MN^{-1})^T=(-N^{-1})(-M)=N^{-1}M.$

   Since $MN=NM$, and both are nonsingular, inverses also commute appropriately, so $N^{-1}M=MN^{-1}.$

5. Substitute into the expression

   $E=M^2N^2(M^TN)^{-1}(MN^{-1})^T$ $=M^2N^2\big[-(MN)^{-1}\big](N^{-1}M).$

   So, $E=-M^2N^2(MN)^{-1}N^{-1}M.$

6. Use commutativity of $M$ and $N$

   Since $MN=NM$, $(MN)^{-1}=N^{-1}M^{-1}=M^{-1}N^{-1}.$

   Therefore, $E=-M^2N^2(N^{-1}M^{-1})N^{-1}M.$

   Rearranging using commutativity, $E=-M^2(M^{-1})N^2(N^{-1})(N^{-1})M$ $=-M\,N\,N^{-1}M$ $=-M^2.$

7. Match with options

   $E=-M^2,$ which corresponds to Option C.

Verification with stored answer

Stored correct answer: C

My derived answer: C

So, the answer agrees with the stored correct answer.