Checking if a $1$-form $\omega$ is closed

Question

I must verify if the $1$-form $\omega = x^2 y dx -3xy^2 dy$ is closed. My computation is as follows: \begin{align} d\omega &= d(x^2 y)\wedge dx + d(-3xy^2)\wedge dy\\ &= (2xy\; dx + x^2 dy)\wedge dx + (-3y^2 \; dx - 6xy\; dy)\wedge dy\\ &= x^2\; dy\wedge dx - 3y^2\; dx\wedge dy\\ &= (-x^2-3y^2)dx\wedge dy \neq 0 \end{align} Hence, $\omega$ is not closed. I don't think I made any mistakes; is this calculation ok?