If $A=\left[\begin{array}{ccc}0& a+2& 2b+1\\ a-1& 0& c-1\\ b-1& 3+c& 0\end{array}\right]$ is a skew-symmetric matrix then $a+b+c=$

$$\begin{gathered} A=-A^T \\ \left[\begin{array}{ccc}0& a+2& 2b+1\\ a-1& 0& c-1\\ b-1& 3+c& 0\end{array}\right]=\left[\begin{array}{ccc}0& 1-a& 1-b\\ -a-2& 0& -3-c\\ -2b-1& 1-c& 0\end{array}\right] \end{gathered}$$

Since $a+2=1-a,2b+1=1-b,3+c=1-c$ 

             $2a=-1$          $3b=0$             $2c=-2$               

               $a=-\frac{1}{2}$           $b=0$                  $c=-1$

                      $a+b+c=-\frac{1}{2}+0-1=-\frac{3}{2}$