How to compute exterior derivative of generalized angular form in $\mathbb R^n$

Question

On $\mathbb R^n$, with $m\in\mathbb Z$, I want to calculate the exterior derivative of the angular form \begin{align} w=\sum_{i=1}^n(-1)^{i-1}\frac{x_i}{\|x\|^m}dx_1\cdots dx_{i-1}\hspace{0.03cm}\widehat{dx_{i}}\hspace{0.03cm}dx_{i+1}\cdots dx_n \end{align} This is the exercise 6 page 261 from the book Analysis on Manifolds of Munkres. First of all, I have no clue what is the meaning of the symbol $\widehat{dx_i}$.Though, I have tried out and got \begin{align} dw&=d\left(\sum_{i=1}^n(-1)^{i-1}\frac{x_i}{\| x \|^m}dx_1\cdots dx_{i-1}\hspace{0.03cm}\widehat{dx_{i}}\hspace{0.03cm}dx_{i+1}\cdots dx_n \right) \newline &=\sum_{i=1}^nd\left[(-1)^{i-1}\frac{x_i}{\|x\|^m}\right]dx_1\cdots dx_{i-1}\hspace{0.03cm}\widehat{dx_{i}}\hspace{0.03cm}dx_{i+1}\cdots dx_n\tag1 \end{align} For each $i=1,\dots n$, we have \begin{align} d\left[(-1)^{i-1}\frac{x_i}{\|x\|^m}\right]&=(-1)^{i-1}\sum_{j=1}^n\partial_j\left(\frac{x_i}{\|x\|^m}\right)dx_j \newline &=(-1)^{i-1}\sum_{\substack{j=1\\ j\ne i}}^n\frac{-mx_ix_j}{\|x\|^{m+2}}dx_j+(-1)^{i-1}\frac{1-mx_i^2\|x\|^{-2}}{\|x\|^m}dx_i\tag2 \end{align} But here I don't know how complicated the equation (1) would be if I substituted (2) into (1). Is there any simple way for me to do this ? Thanks