I found the exerpt below in the book "Introduction to Symplectic Topology". I understand that a 2-form $\omega(p,\zeta, \zeta')$ on $\mathbb R^{2n}$ is a function that maps $\mathbb R^{2n}\times\mathbb R^{2n}\times\mathbb R^{2n}$ to $\mathbb R$. If $\omega$ is independent of $p$, then it essentially maps $\mathbb R^{2n}\times\mathbb R^{2n}$ to $\mathbb R$. That's why I can understand
$$\omega_0(\zeta,\zeta')=\sum_{j=1}^n (\xi_j \eta_j' - \eta_j\xi_j')$$
But what is
$$\omega_0 =\sum_{j=1}^n dx_j\land dy_j$$ and why is it equivalent to the above? What exactly do they mean with $dx_j$ and $dx_j\land dy_j$ here?
I should note that the first $n$ components of $\zeta$ are denoted $\xi$ or $x$ and the last $n$ components of $\zeta$ are denoted $\eta$ or $y$.
The symplectic form
Consider the $2$-form $$\omega_0:=\sum_{j=1}^n dx_j\wedge dy_j\tag{1.1.20}$$ on $\mathbb{R}^{2n}$. We can think of this either as a differential form on $\mathbb{R}^{2n}$ with constant coefficients, or as a nondegenerate skew-symmetric bilinear form $$\omega_0:\mathbb{R}^{2n}\times\mathbb{R}^{2n}\longrightarrow\mathbb{R}$$ on the vector space $\mathbb{R}^{2n}$. (The two notions coincide when the vector space $\mathbb{R}^{2n}$ is viewed as the tangent space of the manifold $\mathbb{R}^{2n}$ at a point $z$.) The value of $\omega_0$ on a pair of vectors $\zeta=(\xi,\eta)$ and $\zeta'=(\xi',\eta')$ with $\xi,\eta,\xi',\eta'\in\mathbb{R}$ is given by $$\omega_0(\zeta,\zeta')=\sum_{j=1}^n(\xi_j\eta_j'-\eta_j\xi_j')=\langle J_0\zeta,\zeta'\rangle=-\zeta^TJ_0\zeta'.\tag{1.1.21}$$
If we define
$$ dx_j(\zeta) = \xi_j $$ $$ dy_j(\zeta) = \eta_j $$
and we define the exterior product like wikipedia:
$$ (f\wedge g)(v_1,\ldots, v_{k+\ell})=\frac{1}{k!\ell!}\sum_{\sigma\in S_{k+\ell}} (sgn(\sigma)) f(v_{\sigma(1)}, \ldots, v_{\sigma(k)})g(v_{\sigma(k+1)} ,\ldots,v_{\sigma(k+\ell)}) $$
then
$$\omega_0(\zeta, \zeta') =\sum_{j=1}^n (dx_j\land dy_j)(\zeta, \zeta') = \sum_{j=1}^n (dx_j(\zeta) dy_j(\zeta') - dx_j(\zeta') dy_j(\zeta)) = \sum_{j=1}^n (\xi_j \eta_j' - \eta_j\xi_j') $$