In Lee's Riemannian Manifolds book, we see that a Riemannian metric $g$ can be expressed locally in coordinates $(x^1, \ldots, x^n)$ by $g = g_{ij} dx^i \otimes dx^j$. Introducing the symmetric product of $1$-forms as $\omega \eta = \frac{1}{2} (\omega \otimes \eta + \eta \otimes \omega)$, we can see that expression is equivalent to $g = g_{ij} dx^i dx^j$.
My question, however, is that I usually interpret $dx^i dx^j$ as the $2$-form $dx^i \wedge dx^j$. As far as I can tell these things are not the same, because according to my computation, for $1$-forms $\omega, \eta$ we'd have $\omega \wedge \eta = \omega \otimes \eta - \eta \otimes \omega$. Is there really such an ambiguity in this notation?
It's very common to write symmetric products using juxtaposition with no product symbol. But you should never omit the wedge symbol in a wedge product. Maybe you're getting confused with the notation for integrals, where, for example $\int f\,dx^1\wedge dx^2$ is defined to mean $\int f(x^1,x^2)\, dx^1dx^2$. The $dx^1dx^2$ in the latter expression is not a $2$-form; it's just part of the notation for integration.