Let $C^{2}(\mathbb{R}^{3};\mathbb{R}^{3})$ be denote the vector space of all skew-symmetric bilinear maps from $\mathbb{R}^{3}\times\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ and let $\operatorname{Lie}(\mathbb{R}^{3})$ be the algebraic subset of $C^{2}(\mathbb{R}^{3};\mathbb{R}^{3})$ consisting of all skew-symmetric bilinear maps $\mu \in C^{2}(\mathbb{R}^{3};\mathbb{R}^{3})$ such that $(\mathbb{R}^3,\mu)$ is a real Lie algebra. The general linear group $\operatorname{GL}(3, \mathbb{R})$ acts by changing of basis on $\operatorname{Lie}(\mathbb{R}^{3})$: if $g\in \operatorname{GL}(3, \mathbb{R})$ and $\mu \in\operatorname{Lie}(\mathbb{R}^{3})$, $g\cdot \mu(x,y) = g\mu(g^{-1}x,g^{-1}y)$, for any $x,y \in \mathbb{R}^{3}$.
I know that the $\operatorname{GL}(3, \mathbb{R})$-orbit of the usual cross product on $\mathbb{R}^3$ is a open set in $\operatorname{Lie}(\mathbb{R}^{3})$ with respect to the subspace topology inherited from the Euclidean topology of $C^{2}(\mathbb{R}^{3};\mathbb{R}^{3})$; for instance, by using the Killing form of $\mathfrak{so}(3,\mathbb{R}) = (\mathbb{R}^3,\times)$. Recall that the cross product $\times$ is determined by $e_1 \times e_2 = e_3$, $e_2 \times e_3 = e_1$, $e_3 \times e_1 = e_2$ and $e_1\times e_1= e_2\times e_2=e_3\times e_3=0$.
I would like to learn/know if the above $\operatorname{GL}(3, \mathbb{R})$-orbit is also a Zariski open set of the algebraic set $\operatorname{Lie}(\mathbb{R}^{3})$, and in such case, what are the polynomials that vanish on the Zariski closure of such orbit in $\operatorname{Lie}(\mathbb{R}^{3})$?
The cross product Lie algebra is the simple Lie algebra $L=\mathfrak{so}_3(\Bbb R)$, which is "formally rigid", because by the second Whitehead Lemma we have $H^2(L,L)=0$. By a result of Gerstenhaber, $L$ then is also "geometrically rigid", which just means that the $GL_3(\Bbb R)$-orbit of it is Zariski open in the whole variety of $3$-dimesnional real Lie algebras. In particular the Zariski closure of this orbit is an irreducible component of the variety. So the answer to your question is positive. It is a good exercise to find the polynomials in the structure constants (in $\Bbb R^{27}$) for $L$ that vanish on the orbit of $L$.