Given any $3\times 3$ finite set of symmetric matrices $A_i$ and positive real $a_i$ such that $\sum_ia_i=1.$ Is there any equivalent condition to the existence of skew symmetric matrices $X_i$ such that:$\DeclareMathOperator{\cof}{cof}$ $$ \cof\left(\sum_{i=1}^na_i(A_i+X_i)\right)=\sum_{i=1}^na_i\cof (A_i +X_i) $$ and $$ \det\left(\sum_{i=1}^na_i(A_i+X_i)\right )=\sum_{i=1}^na_i\det (A_i+X_i). $$ I know that having $\cof\left(\sum_{i=1}^na_iA_i\right)-\sum_{i=1}^na_i\cof A_i $ positive semidefinite is a necessary condition. Is it sufficient?? Or which condition we should add?