If $u_1,\ldots,u_k$ is an orthogonal basis of $W$, then $P_Wv=\sum_i u_i\langle u_i,v\rangle/\|u_i\|^2$. These coefficients are coordinates in that orthogonal basis. With orthonormal columns $Q$, the coefficient vector is $Q^*v$ and the projection in ambient coordinates is $QQ^*v$.
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