Definition 6.1.1 ($k$-form on $\R^n$). A $k$-form on $\R^n$ is a function $\varphi$ that takes $k$ vectors in $\R^n$ and returns a number $\varphi(\vec v_1, \dots, \vec v_k)$ such that $\varphi$ is a multilinear and antisymmetric as a function of the vectors.
Example 6.1.2. Let $i_1, \dots, i_k$ be any $k$ integers between 1 and $n$. Then $dx_{i_1}\land \dots \land dx_{i_k}$ is that function of $k$ vectors $\vec v_1, \dots, \vec v_k$ in $\R^n$ that puts these vectors side by side, making the $n\times k$ matrix
$$ \begin{bmatrix}v_{1,1} & & v_{1, k}\\ \vdots & & \vdots \\ v_{n, 1} && v_{n, k}\end{bmatrix} $$
and selects $k$ rows: first row $i_1$, then row $i_2$, etc. and finally row $i_k$, making a square $k\times k$ matrix, and finally takes its determinant. For instance,
$$ dx_1\land dx_2 \Bigg(\begin{bmatrix}1\\2\\-1\\1\end{bmatrix}, \begin{bmatrix}3\\-2\\1\\2\end{bmatrix}\Bigg) = \det \begin{bmatrix}1 & 3\\ 2 & -2\end{bmatrix} = -8. $$
$$ dx_1 \land dx_2 \land dx_4 \Bigg(\begin{bmatrix} 1\\2\\-1\\1 \end{bmatrix}, \begin{bmatrix} 3\\-2\\1\\2\end{bmatrix} , \begin{bmatrix}0\\1\\2\\1\end{bmatrix} \Bigg) = \det \begin{bmatrix} 1 & 3 & 0\\ 2 & -1 & 1\\ 1 & 2 & 1 \end{bmatrix} = -7. $$
Example 6.1.3 (0-form). Definition 6.1.1 makes sense even if $k=0$. A 0-form on $\R^n$ takes no vectors and returns a number. In other words, it is that number.
<aside> 💡 There are no nonzero $k$-forms on $\R^n$ when $k>n$. If $\vec v_1, \dots, \vec v_k$ are vectors in $\R^n$, and $k > n$, then the vectors are not linearly independent, and at least one of them is a linear combination of the others. Therefore, by antisymmetry, the $k$-form evaluates to 0.
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Definition 6.1.4 (Elementary $k$-forms on $\R^n$). An elementary $k$-form on $\R^n$ is an expression of the form $dx_{i_1}\wedge \dots \wedge dx_{i_k}$, where $1\leq i_1 < \dots < i_k \leq n$. Evaluated on the vectors $\vec v_1, \dots, \vec v_k$, it gives the determinant of the $k\times k$ matrix obtained by selecting rows $i_1, \dots, i_k$ of the matrix whose columns are the vectors $\vec v_1, \dots, \vec v_k$. The only elementary 0-form is the form, denoted 1, which is evaluated on zero vectors.
<aside> 💡 By the pigeon hole principle, there are no elementary $k$-forms when $$k > $$.
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Definition 6.1.5 (Addition of $k$-forms). Let $\varphi$ and $\psi$ be two $k$-forms. Then
$$ (\varphi + \psi)(\vec v_1, \dots\vec v_n) = \varphi(\vec v_1, \dots, \vec v_k) + \psi (\vec v_1, \dots,\vec v_k). $$
Definition 6.1.6 (Multiplication of $k$-forms by scalars).
$$ (a\varphi)(\vec v_1, \dots, \vec v_k) = a(\varphi(\vec v_1, \dots, \vec v_k)). $$