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?
$\omega$ 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}