I am currently reading the book Characteristic classes by Milnor and Stasheff. In chapter 9 they define the euler class of an oriented vector bundle using the Thom isomorphism theorem. In chapter 11 they prove the following (corollary 11.12):
Let $M$ be a compact $n$-dim. smooth oriented manifold and $K$ the field of coefficients. Then it holds $$ \langle e(TM),[M]\rangle=\chi(M) \quad\text{in $K$},$$ where $e(TM)\in H^{n}(M;K)$ is the euler class of $TM$ and $[M]$ is the fundamental class.
In particular, when $K=\mathbb{R}$ and using the isomorphism $$ H^{n}(M;\mathbb{R})\cong H_{dR}^{n}(M) $$ this becomes $$ \int_M \omega_{TM}=\chi(M), $$ where $\omega_{TM}$ corresponds to $e(TM)$ under the above isomorphism. As this looked dangerously close to the Gauss-Bonnet Theorem, I tried to look up if the Gauss-Bonnet Theorem can be obtained from the above. I found that the euler class can also be defined in terms of differential forms and that the above result still holds. However, I have a problem with the definition using differential forms. $$\text{I don't see how this is related to the definition using the Thom class.} $$ To explain more precisely what I don't understand I will quickly go over the definition using the Thom class. The fact that $M$ is a smooth oriented manifold means that we have a choice of generator $\mu^x$of $H^n(T_xM,T_xM\setminus\{0\})$ that is locally consistent. Then by the Thom isomorphism theorem there exists a unique class $\mu \in H^n(TM,TM\setminus M\times \{0\})$ s.t. the restriction of $\mu$ to each $H^n(T_xM,T_xM\setminus\{0\})$ equals $\mu^x$. Then the euler class $e(TM)$ is defined as the image of $\mu$ under the composition $$ H^n(TM,TM\setminus M\times \{0\})\to H^n(TM)\xrightarrow{s^*} H^n(M), $$ where $s:M\to TM$ is the zero section.
As said before, I want to relate this to the Gauss-Bonnet Theorem. I am particularly interested in the case $n=2$. So I would like to prove that the differential form $\omega_{TM}$ corresponding to $e(TM)$ is $$ \omega_{TM}=\frac{1}{2\pi}K\omega_g, $$ where $K$ is the Sectional/Gaussian curvature and $\omega_g$ is the Riemannian volume form.
I know that any orientation form $\omega$ on $M$, i.e. any nowhere vanishing 2-form on $M$, encodes a locally consistent choice of orientation for $T_xM$. My problem is however that I do not know how to describe $H^n(TM,TM\setminus M\times \{0\})$ using differential forms.
To sum up, I would be grateful for clarifications how the "Thom class-definition" and the "differential form-definition" of the euler class relate to each other (in general or in dimension 2) or a direct way to see how the stated result from Characteristic Classes implies the Gauss-Bonnet Theorem (in dimension 2).