Differential Form
Definition. A $k$-form $\omega \in \Omega^k(M)$ smoothly assigns to each $p$ an antisymmetric multilinear map $\omega_p: (T_pM)^k \to \mathbb{R}$. In coordinates: $\omega = \frac{1}{k!}\omega_{i_1\dots i_k} dx^{i_1} \wedge \cdots \wedge dx^{i_k}$ with totally antisymmetric components. The wedge product $\wedge$ is the antisymmetrized tensor product; $\alpha \wedge \beta = (-1)^{|\alpha||\beta|}\beta \wedge \alpha$.
Intuition. A $k$-form is the thing you integrate over a $k$-dimensional surface: 1-forms over curves ($\int A_i dx^i$), 2-forms over surfaces (flux), $n$-forms over regions (volume).
Physics anchor. Landau's "antisymmetric 4-tensor" $F_{ik}$ (§23) is the component notation for the 2-form $F = \frac{1}{2}F_{ik}\,dx^i \wedge dx^k$. The four-potential $A_i$ is the 1-form $A = A_i dx^i$. Antisymmetry isn't an accident of EM — it's what makes these objects integrable over worldsheets and surfaces.
Related: Exterior Derivative · Stokes Theorem · de Rham Cohomology · Pushforward and Pullback · Connection 1-Form · Module 02 - Differential Forms Refresher
Sources: Sternberg Ch. I, III; Landau §6 (antisymmetric tensors), §23.