Formula for the exterior derivative of a p-form Let $$\omega = \sum_{i_1 < \dots < i_p} f_{i_1\dots i_p} dx_{i_1} \wedge \dots \wedge dx_{i_p}$$ be a $p$-form over $\mathbb{R}^n$. I am interested in a formula for the exterior derivative $d\omega$. All I could find was this answer for $p=2$:
$$\omega = \sum_{i < j} f_{ij} dx_i\wedge dx_j \quad \Rightarrow \quad d\omega = \sum_{i<j<k} \left( \frac{\partial f_{jk}}{\partial x_i} - \frac{\partial f_{ik}}{\partial x_j} + \frac{\partial f}{\partial x_k} \right) dx_i \wedge dx_j \wedge dx_k$$
Let $\omega = \left(z^2 - x^2\right)dx \wedge dy + \left(z^2 - x^2\right)dx \wedge dz$ be a 2-form over $\mathbb{R}^3$ as an example, then:
\begin{align}
d\omega = &\left( - \frac{\partial}{\partial y} \left( z^2 - x^2 \right) + \frac{\partial}{\partial z} \left( z^2 - x^2 \right) \right) dx \wedge dy \wedge dz\\
= &\ 2zdx \wedge dy \wedge dz
\end{align}
What is the formula for the general case?
 A: $\omega$ is a $p$-form over $\mathbb{R}^3$ in each example.
$p=0$:
The following formula holds by definition and can't be simplified further:
\begin{align}
\omega =\ &f\\
\Rightarrow d\omega =\ &\sum_i \frac{\partial f}{\partial x_i} dx_i\\\\
\omega =\ &xy - xz + z\\
\Rightarrow d\omega =\ &\frac{\partial}{\partial x}\left(xy - xz + z\right)dx + \frac{\partial}{\partial y}\left(xy - xz + z\right)dy + \frac{\partial}{\partial z}\left(xy - xz + z\right)dz\\
=\ &\left( y - z \right)dx + xdy + (1-x)dz
\end{align}
$p=1$:
Remember that $dx_i \wedge dx_j = - dx_j \wedge dx_i$ and consequently $dx_i \wedge dx_i = 0$. The following formula holds by definition:
\begin{align}
\omega =\ &\sum_{i} f_i dx_i\\
\Rightarrow d\omega =\ &\sum_i \sum_j \frac{\partial f_i}{\partial x_j} dx_j \wedge dx_i\\\\
\omega =\ &+ (x-y)dx\\
&+ zdy\\
&+ (1-z)dz\\\\
\Rightarrow d\omega =\ &+\frac{\partial}{\partial x}(x-y)dx\wedge dx &&+ \frac{\partial}{\partial y}(x-y)dy\wedge dx &&+ \frac{\partial}{\partial z}(x-y)dz\wedge dx\\
&+ \frac{\partial}{\partial x}(z)dx\wedge dy &&+ \frac{\partial}{\partial y}(z)dy\wedge dy &&+ \frac{\partial}{\partial z}(z)dz\wedge dy\\
&+ \frac{\partial}{\partial x}(1-z)dx\wedge dz &&+ \frac{\partial}{\partial y}(1-z)dy\wedge dz &&+ \frac{\partial}{\partial z}(1-z)dz\wedge dz\\\\
=\ &+0 &&+ \frac{\partial}{\partial y}(x-y)dy\wedge dx &&+ \frac{\partial}{\partial z}(x-y)dz\wedge dx\\
&+ \frac{\partial}{\partial x}(z)dx\wedge dy &&+0 &&+ \frac{\partial}{\partial z}(z)dz\wedge dy\\
&+ \frac{\partial}{\partial x}(1-z)dx\wedge dz &&+ \frac{\partial}{\partial y}(1-z)dy\wedge dz &&+0\\\\
=\ &+ \frac{\partial}{\partial x}(z)dx\wedge dy &&+ \frac{\partial}{\partial y}(x-y)dy\wedge dx\\
&+ \frac{\partial}{\partial x}(1-z)dx\wedge dz &&+ \frac{\partial}{\partial z}(x-y)dz\wedge dx\\
&+ \frac{\partial}{\partial y}(1-z)dy\wedge dz &&+ \frac{\partial}{\partial z}(z)dz\wedge dy\\\\
=\ &+ \frac{\partial}{\partial x}(z)dx\wedge dy &&- \frac{\partial}{\partial y}(x-y)dx\wedge dy\\
&+ \frac{\partial}{\partial x}(1-z)dx\wedge dz &&- \frac{\partial}{\partial z}(x-y)dx\wedge dz\\
&+ \frac{\partial}{\partial y}(1-z)dy\wedge dz &&- \frac{\partial}{\partial z}(z)dy\wedge dz
\end{align}
\begin{align}
=\ &+ \left( \frac{\partial}{\partial x}(z) - \frac{\partial}{\partial y}(x-y) \right) dx\wedge dy\\
&+ \left( \frac{\partial}{\partial x}(1-z) - \frac{\partial}{\partial z}(x-y) \right) dx\wedge dz\\
&+ \left( \frac{\partial}{\partial y}(1-z) - \frac{\partial}{\partial z}(z) \right) dy\wedge dz\\\\
=\ &- dx\wedge dy\\
&+ 0\\
&+ dy\wedge dz\\\\
=\ &dy\wedge dz - dx\wedge dy
\end{align}
This formula can be simplified:
\begin{align}
\omega =\ &\sum_i f_i dx_i\\
\Rightarrow d\omega =\ &\sum_i \sum_j \frac{\partial f_i}{\partial x_j} dx_j \wedge dx_i\\
=\ &\sum_{i < j} \left( \frac{\partial f_j}{\partial x_i} - \frac{\partial f_i}{\partial x_j} \right) dx_i \wedge dx_j
\end{align}
Notice how applying the simplified formula skips nearly all the steps in the computation above.
$p=2$:
\begin{align}
\omega =\ &\sum_{i < j} f_{ij} dx_i\wedge dx_j\\
\Rightarrow d\omega =\ &\sum_{i < j} \sum_k \frac{\partial f_{ij}}{\partial x_k} dx_k \wedge dx_i \wedge dx_j\\
=\ &\sum_{i<j<k} \left( \frac{\partial f_{jk}}{\partial x_i} - \frac{\partial f_{ik}}{\partial x_j} + \frac{\partial f_{ij}}{\partial x_k} \right) dx_i \wedge dx_j \wedge dx_k
\end{align}
$p=3$:
\begin{align}
\omega =\ &\sum_{i < j < k} f_{ijk} dx_i \wedge dx_j \wedge dx_k\\
\Rightarrow d\omega =\ &\sum_{i < j < k} \sum_l \frac{\partial f_{ijk}}{\partial x_l} dx_l \wedge dx_i \wedge dx_j \wedge dx_k\\
=\ &\sum_{i<j<k<l} \left( \frac{\partial f_{jkl}}{\partial x_i} - \frac{\partial f_{ikl}}{\partial x_j} + \frac{\partial f_{ijl}}{\partial x_k} - \frac{\partial f_{ijk}}{\partial x_l} \right) dx_i \wedge dx_j \wedge dx_k \wedge x_l
\end{align}
$p \in \mathbb{N}_0$:
\begin{align}
\omega =\ &\sum_{i_1 < \dots < i_p} f_{i_1\dots i_p} dx_{i_1} \wedge \dots \wedge dx_{i_p}\\
\Rightarrow d\omega =\ &\sum_{i_1 < \dots < i_p} \sum_j \frac{\partial f_{i_1\dots i_p}}{\partial x_j} dx_j \wedge dx_{i_1} \wedge \dots \wedge dx_{i_p}\\
=\ &\sum_{i_1 < \dots < i_{p+1}} \left( \sum_{j=1}^{p+1} (-1)^{j-1} \frac{\partial f_{i_1\dots i_{j-1}i_{j+1}\dots i_{p+1}}}{\partial x_{i_j}} \right) dx_{i_1} \wedge \dots \wedge dx_{i_{p+1}}
\end{align}
