  We introduce $\varepsilon$-approximate versions of the notion of Euclidean
vector bundle for $\varepsilon \geq 0$, which recover the classical notion of
Euclidean vector bundle when $\varepsilon = 0$. In particular, we study
\v{C}ech cochains with coefficients in the orthogonal group that satisfy an
approximate cocycle condition. We show that $\varepsilon$-approximate vector
bundles can be used to represent classical vector bundles when $\varepsilon >
0$ is sufficiently small. We also introduce distances between approximate
vector bundles and use them to prove that sufficiently similar approximate
vector bundles represent the same classical vector bundle. This gives a way of
specifying vector bundles over finite simplicial complexes using a finite
amount of data, and also allows for some tolerance to noise when working with
vector bundles in an applied setting. As an example, we prove a reconstruction
theorem for vector bundles from finite samples. We give algorithms for the
effective computation of low-dimensional characteristic classes of vector
bundles directly from discrete and approximate representations and illustrate
the usage of these algorithms with computational examples.
