Prove that the set of skew-symmetric matrices is closed under addition
Let $A,B\in W$. Then $$-(A+B)^T=-(A^T+B^T)=-A^T-B^T=A+B$$ The final equality comes from $A=-A^T$ and $B=-B^T$ (because they are in $W$). So $A+B=-(A+B)^T$ so $A+B\in W$.
Prove that the set of skew-symmetric matrices is closed under addition
Let $A,B\in W$. Then $$-(A+B)^T=-(A^T+B^T)=-A^T-B^T=A+B$$ The final equality comes from $A=-A^T$ and $B=-B^T$ (because they are in $W$). So $A+B=-(A+B)^T$ so $A+B\in W$.