Idea
I have an idea, that it is possible to generalize the winding number for surfaces of the form $f: \mathbb{R}^n \rightarrow \mathbb{R}^{n+1}$
The winding number for $n=1$ is $w_{\gamma}(x) = \oint_{\gamma} \frac{\left < L r, \dot r\right >}{|r|^2} dt$, where $L = \begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}$ and $r = \gamma - x$.
I would say that $w_{\gamma}$ is similar to this calculation:
$$\hat{w}_{\gamma}(x) = \oint_{\gamma}\left < \frac{d}{dt} \frac{r}{|r|}, L \frac{r}{|r|}\right > dt$$
We can think of $\frac{r}{|r|}$ as a projection of the curve $\gamma$ to a unit circle centered at $x$. The $\frac{d}{dt} \frac{r}{|r|}$ can be thought as the tangent vector of the projection to this unit circle. The value $\left |\frac{d}{dt} \frac{r}{|r|}\right |$ is the infinitesimal angle of this projection.
So $\hat{w}_{\gamma}(x)$ can be though as the signed angle.
Questions
Does $\hat{w}_{\gamma}(x)$ has the same meaning as $w_{\gamma}(x)$?
Can we apply the same approach of $\hat{w}_{\gamma}(x)$ to generalize to surfaces $f: \mathbb{R}^n \rightarrow \mathbb{R}^{n+1}$?
What is the main approach among mathematicians to generalize the winding number?
First question attempt
I tried to change the form of $\hat{w}_{\gamma}(x)$:
$$\hat{w}_{\gamma}(x) = \oint_{\gamma}\left < \frac{d}{dt} \frac{r}{|r|}, L \frac{r}{|r|}\right > = \oint_{\gamma}\frac{1}{|r|^2}\left < \text{ort}_{r}\dot r, L \frac{r}{|r|}\right > = \oint_{\gamma}\frac{1}{|r|^4}\left (\left < \dot r, L r\right > |r|^2 - \left < r, \dot{r} \right >\left < r, Lr\right >\right )$$
$$\hat{w}_{\gamma}(x) = w_{\gamma}(x) - \oint_{\gamma}\frac{1}{|r|^4}\left< r, \dot r \right>\left< r, Lr\right>\;\;(1)$$
Since $r$ is orthogonal to $Lr$ then $\left< r, Lr\right> = 0$.
$$(1) = \hat{w}_{\gamma}(x) = w_{\gamma}(x) $$
Second question attempt
I tried to generalize to $n=2$:
Let $\gamma: U \in \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be the parameterization of a closed surface in $\mathbb{R}^3$.
Then $\hat{w}_{\gamma}(x)$ will be:
$$\hat{w}_{\gamma}(x) = \iint_{\gamma} \star\hat r^{\flat} \left(\partial_{u_1} \hat r, \partial_{u_2} \hat r \right)$$
Where is $\star$ is the Hodge star operator, $\flat$ is the flat musical isomorphism, $\hat r = \frac{r}{|r|}$ and $\partial_{u_k} f = \frac{\partial f}{\partial u_k}$.
I am not a mathematician so please correct anything that is wrong, especially the differential forms parts.
Thanks in advance.