Orthogonal Projection of a matrix Let $V$ be the real vector space of $3 \times 3$ matrices with the bilinear form $\langle A,B \rangle=$ trace $A^tB$, and let $W$ be the subspace of skew-symmetric matrices. Compute the orthogonal projection to $W$ with respect to this form, of the matrix $$\begin{pmatrix} 1& 2 & 0\\ 0 & 0 & 1\\ 1 & 3 & 0\end{pmatrix}$$
Could someone show me how to proceed ?

Find the orthogonal complement to $W$, i.e. 
$$W^{\perp} := \{ X \in V : \langle X, W \rangle = 0\}$$
Assuming that $\dim W + \dim W^{\top} = \dim V$, we can write any $X \in V$ as a linear combination:
$$X = \alpha A + \beta B$$
where $\alpha,\beta \in \mathbb{R}$, $A \in W$ and $B \in W^{\top}$. The orthogonal projection $\pi : V \twoheadrightarrow  W$ is given by
$$\pi : \alpha A + \beta B \longmapsto  \alpha A$$
