Fix a basis $\{c_1,c_2, \dots, c_n\}$ of $C$. Since $\mathbb{R}^{2n}=A\oplus B$, for each $i \in \{1,2, \dots, n\}$, there exist $a_i\in A$, $b_i\in B$ such that $c_i=a_i+b_i$. Now, $a_1,a_2, \dots, a_n$ are linearly independent and hence a basis of $A$. Indeed, suppose $\sum\lambda_ia_i=0$, then $\sum\lambda_ic_i=\sum\lambda_ib_i\in C\cap B$, hence $\sum\lambda_ic_i=0$ and so all $\lambda_i=0$. In the same way, $b_1,b_2, \dots, b_n$ are a basis of $B$.