Degree of a map $\phi: R^{n} \rightarrow S^{n}$

Question

I've read a few papers in which they state that the winding number of a mapping $\phi: R^{3} \rightarrow S^{3}$ can be written as the integral $$\int_{\mathcal{R}^3} \epsilon_{ijk}\epsilon^{abc} \partial_a \phi^i \partial_b \phi^j \partial_c \phi^k\ dx.$$ Does anyone know if there is a similar formula for the winding number in the case of a mapping $R^{n} \rightarrow S^{n}$?