Is there an algebraic version of Darboux's theorem?

Let $M$ denote a smooth manifold, and $omega in Omega^2(M, mathbb{R})$ a symplectic form. The classical version of Darboux’s theorem states that for any $x in M$, there exists an open neighborhood $U$ of $x$ together with local coordinates $p_1,dots, p_n, q_1, dots, q_n$ on $U$ such that
begin{equation}
omega vert_U = dp_1 wedge dq_1 + dots + dp_n wedge dq_n. tag{1}
end{equation}
I learned from this question that the theorem also holds in the complex analytic context, which leads to my question. All of the ingredients in Darboux’s theorem make sense algebraically. Specifically, let $X$ denote a smooth variety over an algebraically closed field $k$ of characteristic $0$. A symplectic form on $X$ is a closed 2-form $omega in H^0(X, Omega^2_{X/k})$ inducing a non-degenerate pairing on the tangent space to each closed point of $X$. For example, we have the ‘standard’ symplectic form on $mathbb{A}^{2n}_k = text{Spec} , k[p_1,dots, p_n, q_1, dots, q_n]$ given by (1).

My question, vaguely stated, is whether or not there is a version of Darboux’s theorem for such symplectic forms.

For example, since $X$ is smooth, there exists an open neighborhood $U$ of $x$, and an étale $k$-morphism $f: U rightarrow mathbb{A}^n_k$, where $n = text{dim}(X)$. Perhaps such a result would state that $n = 2m$ is even, and that $f$ can be chosen so that $omega$ is the pullback of the standard form on $mathbb{A}^{2m}_k$ under $f$.