Well, the Euler class exists as an obstruction, as with most of these cohomology classes. It measures "how twisted" the vector bundle is, which is detected by a failure to be able to coherently choose "polar coordinates" on trivializations of the vector bundle.

In the case where $E = \mathbb{R}^2 \times B$ is a trivial vector bundle with projection map $\pi: E \rightarrow B$, then we can take the form $\phi$ to be the pullback of the form $\frac{1}{2\pi} d \theta$ under the projection $$E - E_0 = (\mathbb{R}^2 - 0) \times B \rightarrow \mathbb{R}^2 - 0 $$ where $d\theta$ is the standard angular form on $\mathbb{R}^2 - 0$. Then we have that the Euler class $\chi$ is defined by
$$d \phi = - \pi^* \chi$$ a formula that is generally true, if you think about defining local polar coordinates and having $\phi$ measuring how they fail to piece together over triple intersections.

In this case, $\phi$ is closed and therefore $e$ is 0. This stems from the fact that we can choose global angular coordinates on the vector bundle, which is what this thing measures the failure of in general.

For a thorough discussion of the Euler class from this perspective (and a chance to read a great book that can give you geometric intuition for these algebro-topological things) check out Differential Forms in Algebraic Topology by Bott & Tu.