Here are some more possible approaches to show that the cohomology of $\Omega U(n)$ is torsion-free, as a complement to Neil Strickland's answer: 

In general, a useful tool to compute the cohomology of loop spaces is the *Eilenberg-Moore spectral sequence*, of which you can find an overview in McCleary's book "A user's guide to spectral sequences". It may well be (I do not have my copy handy) that the particular example of $\Omega U(n)$ is discussed in there. 

In the specific case of loop spaces of compact Lie groups, there is a cell decomposition of $\Omega G$ due to Bott, cf. R. Bott: An application of the Morse-theory to the topology of Lie groups", BSMF 84, 251-281. Another approach for the computation of this cell structure using the Bruhat decomposition, cf. H. Garland and M.S. Raghunathan: A Bruhat decomposition for the loop space of a compact group: a new approach to a result of Bott. 
There is a lot more literature on the homotopy type of $\Omega U(n)$ refining cohomology computations. For instance, there is a stable splitting of $\Omega U(n)$ (which refines and is built on Bott's cell structure mentioned above), cf. M. Crabb: On the stable splitting of $U(n)$ and $\Omega U(n)$. LNMA 1298, 35-53. 

There is also a Langlands type approach to determine the homology of loop groups, cf. [Z. Yun and X. Zhu: Integral homology of loop groups via Langlands dual groups][1]; DOI: [10.1090/S1088-4165-2011-00399-X](https://doi.org/10.1090/S1088-4165-2011-00399-X), arXiv: [0909.5487](https://arxiv.org/abs/0909.5487).


  [1]: https://math.mit.edu/~zyun/Homology_Loop_published.pdf