I am looking for a reference that discusses the representation theory of Lie groups, especially the classical groups, on real vector spaces. It seems that almost every text I've looked at restricts only to representations on complex vector spaces, since the theory is easier in the complex case. However, for my purposes I would like to know about real representations of Lie groups, since I am working almost exclusively with real vector bundles. I don't really understand how to find out information about real representations from the complex theory.
In particular, I would like to learn about the irreducible real representations of $O(n, \mathbb{R})$ and $GL(n, \mathbb{R})$, especially on tensors. For instance, I would like to see a proof that a representation of $O(n)$ on covariant $2$-tensors decomposes into trace and trace-free parts etc.
Ideally I would prefer a text with a more categorical point of view, but it's probably asking a bit much.