$\def\RR{\mathbb{R}}\def\Hom{\mathrm{Hom}}$This seems too simple, so please tell me what I'm missing. We first prove the corresponding result in homology. Let $C_i(X)$ be the group of singular $i$-chains in $X$. Then $C_i(M) = \lim_{j \to \infty} C_i(M_j)$ (because $i$-simplices are compact, so the image of any finite number of them must lie in some $M_j$). Homology commutes with direct limits, so we deduce that $H_i(M) = \lim_{j \to \infty} H_i(M_j)$.

Now, the universal coefficient theorem tells us that $H^i(M, \RR) = \Hom(H_i(M), \RR)$ and the same for $M_j$. So the question is whether $\Hom( \ , \RR)$ turns direct limits into inverse limits, and the answer is yes.  

Both the universal coefficient theorem, and the identification of singular and de Rham cohomology, work for general manifolds.