I know that I have to use:
$$\oint F\cdot dr = \iint \operatorname{curl}(F)\cdot n \;dS,$$
with $\operatorname{curl}(F)= -x j -yk$ and $n\;dS = dA\; k$
But the dot product of the curl and $dS$ leaves me with $$\iint -y \; dA$$ but I am unsure how to proceed from here. Do I need to change coordinate systems as I am dealing with a cicle? Any help is appreciated!
In order to apply Stoke's Theorem to evaluate $\int_CF\cdot d\vec{r}$ where $C$ is the circular loop $x^2+y^2=a^2$ you need to first establish a surface $S$ whose boundary is $C$ and is equipped with a normal vector $\vec{n}$ that induces the original orientation prescribed to $C$. Let's take $S$ to be the circular disc $$\{x^2+y^2\leq a^2,z=0\}$$ which we'll parameterize by $$\vec{r}(u,v)=(u\cos(v),u\sin(v),0)$$ on the domain $0\leq u\leq a$ and $0\leq v\leq 2\pi$. Clearly the vector $\vec{n}=\big<0,0,1\big>$ is perpendiular to $S$ and induces a counterclockwise orientation on its boundary $C$. (I'm assuming you're prescribing a counterclockwise orientation to $C$.) A little bit of calculation also reveals that $$\text{curl}(F)=\big<0,-1,-1\big>$$ $$\vec{r}_u \times \vec{r}_v=\big<0,0,u\big>$$ $$dS=||\vec{r}_u \times \vec{r}_v||dudv=ududv$$ So with Stoke's Theorem, $$\int_C F \cdot d\vec{r}=\int_{S}\big[\text{curl}(F)\cdot \vec{n}\big]dS=\int_0^{2\pi} \int_0^a -ududv=-\pi a^2$$