Suppose $V$ is a finite-dimensional vector space with a (linear) complex structure $J$. Further assume that the vectors $u_1,...,u_n$ form a basis of $V$ over $\mathbb{C}$. Then the vectors $u_1,Ju_1,...,u_n,Ju_n$ form a basis of $V$ over $\mathbb{R}$.
(Edit: $J:V\rightarrow V$ is any linear map with $J^2=-id$.)
It is clear that both the sets ${u_1,...,u_n}$ and ${Ju_1,...,Ju_n}$ are linearly independent. But so far I was not able, neither with linear combinations nor with inner products, to show that the whole set of vectors forms a basis.
Suppose that there exist $a_1, \ldots, a_n, b_1, \ldots, b_n$ satisfying $\sum_{k} a_ku_k+\sum_{k} b_kJu_k=0$. Since $u_i$'s are complex basis of $V$, you get $a_k+jb_k=0$ for all $k$.