Does a vector bundle need to be considered in an ambient space?

Question

We may consider manifolds without embedding them into $\mathbb{R}^n$ for example.

Is it possible to do the same with vector bundles, or must they always sit inside $\mathbb{R}^n$?

Answer 1

Yes, it is definitely possible, but you need to work a little.
Start out by letting your manifold have a differentiable structure. This enables you to define smooth (differentiable) functions.
Then define the tangent space at a point by using one of your favourite definitions - they all give the same result. Note that the tangent space at a point is a highly abstract notion - this is essentially the price you pay for not using an ambient space.
Next you take the disjoint union of all tangent spaces at all points of the manifold and give it a manifold structure by explicitly constructing coordinate charts. You then have the so called tangent bundle

Also note that the tangent bundle is indeed a special case of a (smooth) vector bundle or more generally a (smooth) fibre bundle.