Find all the real subspaces $W$ of $V$ satisfying the properties.

Question

Suppose $V=\mathbb C^n$,find all the real subspaces $W$(real subspace means subspace over real field) of $V$ such that for each $\alpha$ in $V$ there exists unique $\beta,\gamma$ in $W$ satisfying $\alpha=\beta+i\gamma$.For sure $\mathbb R^n$ is one of them,but I want to find precisely all of them.If I could do that then with the help of the Lemma stated in Hoffmann Kunze Linear Algebra P-$276$,problem $14$,I can find all conjugations on $\mathbb C^n$ explicitly.I have done it for $n=1$.$W$ is precisely the sets $c\mathbb R$,where $|c|=1,c\in \mathbb C$.But I am having problem with $n>1$.