If $X$ is a quasi-projective variety, then for each integer $n\geq 0$ one can define the symmetric product to be the scheme quotient $$\textrm{Sym}^n(X)=X^n/S_n,$$ where $S_n$ is the symmetric group and $X^n$ is $n$-fold power of copies of $X$ over the base field. I am not sure, but I think one can also characterize $\textrm{Sym}^n(X)$ as the scheme representing the functor of "unordered $n$-tuples of points" in $X$, which I should probably refer to as effective zero cycles of degree $n$.
I heard that the construction of $\textrm{Sym}^n(X)$ does not work if $X$ is not necessarily quasi-projective. I suppose this means (both) that there is no scheme quotient $X^n/S_n$ as above, and that the above functor is not representable.
Question. Is there a geometric object (maybe not a scheme anymore) that replaces $\textrm{Sym}^n(X)$ when $X$ is not quasi-projective?
Thank you in advance.