$$\omega=\sum_{i=1}^n(-1)^{i-1}f_idx_{1}\wedge dx_{i-1}\wedge dx_{i+1}\wedge ...\wedge dx_{n}$$
Where $$f_i= {x_i\over|x|^m}$$
Calculate $d\omega$
Now, I feel like this is the next step to make, but im unsure where to go from here. The textbook explained the differential operator fairly quickly, so im not sure i really understand it. $$d\omega=\sum_{i=1}^n(-1)^{i-1}df_i\wedge dx_{1}\wedge dx_{i-1}\wedge dx_{i+1}\wedge ...\wedge dx_{n}$$
Your next step is correct. Now $$ df_i=\sum_{j=1}^n\frac{\partial f_i}{\partial x_j}dx_j, $$ but as the exterior product is zero for repeated $dx_i$, the only terms that make a contribution for $d\omega$ in the above sum have indices $(2,\ldots ,i-2)$ and $i$.