Tangent bundles of smooth manifolds

Question

Using the identity $T(M \times N) = T(M) \times T(N)$, it is easy to construct the tangent bundles for various smooth manifolds such as the n-dimensional sphere $S^{n}$. However, I could not figure out a way to picture the tangent bundles for more complicated manifolds such as the klein bottle or the real projective plane. For the klein bottle, would one have to use an identity like the above to construct the tangent bundle? More importantly, how would one be able to picture it? I have seen many examples in which tangent bundles may be decomposed into the direct sum of several bundles, would this make it easier to picture the tangent bundle for these manifolds?