Hodge Star Operator Causing Sign Change

I am currently in the process of teaching myself Differential Forms. I have recently begun to learn about the Hodge Star $(\star)$. I was reading about it and came across the following equality for $2$-space: $$ \star\mathrm{d}y = -\mathrm{d}x $$

Just to clarify, I understand that $\star$ converts $k$-forms into $(n-k)$-forms in $n$-space. However, I do not understand where this negative comes from or why it changes to a $\mathrm{d}x$, Why actually is this the case, intuitively?

It's about orientation. $\omega\wedge\star\omega$ has to give a nonnegative multiple of the volume form. In your case, $dx\wedge dy$ is the volume (area) form on $\Bbb R^2$ with the standard orientation, and $dy\wedge (-dx) = dx\wedge dy$. (It may also be useful to check that $\|{\star}\omega\| = \|\omega\|$.)