Example of compactly supported, closed, smooth $2$-form in $\mathbb{R}^3$

Question

I want to find an example of compactly supported closed, smooth $2$-form in $\mathbb{R}^3$. I know that for a $2$-form $$w = a dx\wedge dy + b dx\wedge dz + c dy\wedge dz$$ and its exterior derivative is $$dw= (\frac{\partial a}{\partial z}-\frac{\partial b}{\partial y}+\frac{\partial c}{\partial x})dx\wedge dy\wedge dz$$ So I want to pick a compact supported, smooth function $\phi:\mathbb{R}^3 \to \mathbb{R}$ that is symmetric across $x,y,z$ and choose $\psi(x,y,z)=\phi(x,2y,z)$ such that $2\partial_x\psi= 2\partial_z\psi = \partial_y\psi$. Then choose $$w(p) = \psi(p) dx\wedge dy + \psi(p) dx\wedge dz + \psi(p) dy\wedge dz$$ and by equation above $dw = 0$ so $w$ is closed. I wonder if this is legitimate. The question is that the solution is not symmetric and there is $2y$ for seemingly no reason, so I think I probably did something wrong.