$\left(\begin{array}{ccc}2& 3& 5\\ 4& 1& 2\\ 1& 2& 1\end{array}\right)=P+Q$ where $P$ is a symmetric and $Q$ is a skew-symmetric matrices then $Q$ is 

We know that any square matrix can be expressed as sum of a symmetric and a skew symmetric matrices 

If $A=P+Q$

Where $P$ is symmetric matrix and it can be expressed as $P=\frac{1}{2}\left(A+A^T\right)$ and $Q$ is a skew symmetric matrix and it can be expressed as $Q=\frac{1}{2}\left(A-A^T\right)$

Hence, 

$\begin{array}{rcl}Q& =& \frac{1}{2}\left(A-A^T\right)\\ & =& \frac{1}{2}\left\{\left[\begin{array}{ccc}2& 3& 5\\ 4& 1& 2\\ 1& 2& 1\end{array}\right]-\left[\begin{array}{ccc}2& 4& 1\\ 3& 1& 2\\ 5& 2& 1\end{array}\right]\right\}\\ & =& \frac{1}{2}\left\{\left[\begin{array}{ccc}2-2& 3-4& 5-1\\ 4-3& 1-1& 2-2\\ 1-5& 2-2& 1-1\end{array}\right]\right\}\\ & =& \frac{1}{2}\left[\begin{array}{ccc}0& -1& 4\\ 1& 0& 0\\ -4& 0& 0\end{array}\right]\\ & =& \left[\begin{array}{ccc}0& -\frac{1}{2}& 2\\ \frac{1}{2}& 0& 0\\ -2& 0& 0\end{array}\right]\end{array}$