Degree of continuous mapping via integral

Question

Let $f \in C(S^{n},S^{n})$. If $n=1$ then the degree of $f$ coincides with index of curve $f(S^1)$ with respect to zero (winding number) and may be computed via integral $$ \deg f = \frac{1}{2\pi i} \int\limits_{f(S^1)} \frac{dz}{z} $$ Is it possible to compute the degree of continuous mapping $f$ in the case $n>1$ via integral of some differential form?

Answer 1

You could find some useful information (try page 6) here and here.