Let $V \rightarrow M$ be an oriented vector bundle over a compact oriented manifold $M$ equipped with a metric $h$ (the metric $h$ is a metric on the Vector bundle $V$, not on the manifold $M$). Is there some ``natural'' differential $\omega_T$ form representing the Thom Class of $V$? In particular I want the following properties:

1. If $(M,g)$ is a compact two dimensional Riemannian manifold,
   and $V = TM$, the tangent bundle of $M$ and $X_0 : M \rightarrow TM$ the zero vector field, then $$ X_0^{*} (\omega_T) = \frac{K}{2 \pi} dA $$ equality holding on the level of forms, where $K$ is the Gaussian curvature and $dA$ is the area form.

2. If $V\rightarrow M$ is a complex vector bundle with a hermitian metric $h$ and $s_o : M \rightarrow V$ the zero section then $s_0^*(\omega_T)$ is the differential form for the top Chern class obtained by Chern Weil theory (again equality holds on the level of forms).

Notice that on the level of cohomology, the pull back via the zero section of the Thom class gives us the Euler class of $V$. My basic question is that what should one take the Thom class to be, to obtain equality on ``the level of forms'' when there is a natural form representing the Euler class.