Why are vectors and one-forms treated as equal in Euclidean space? I know that in general a vector space $V$ and its dual space $V^{\ast}$ are distinct from one another, and that given a metric $g$ an isomorphism between the two can be constructed, defined by $V^{\mu}=g^{\mu\nu}\omega_{\nu}$, where $V=V^{\mu}e_{\mu}\in V$ and $\omega =\omega_{\mu}dx^{\mu}\in V^{\ast}$. As I understand ( I may be incorrect), one can use quantities other than the metric to construct such an isomorphism between the two spaces and hence $V$ and $V^{\ast}$ aren't naturally isomorphic, in general. 
However, in (slightly more) elementary linear algebra, when considering Euclidean (flat) space, endowed with the standard Euclidean metric, we treat vectors and one-forms as if they are the same object - one does not distinguish between the two. Why is it possible to do this? Is it simply that in this case there is a natural isomorphism between the two (given via the canonical basis for $\mathbb{R}^{n}$), or is there something else to it?
 A: You are correct that a choice of nondegenerate bilinear form $g$ on a vector space $V$ induces an isomorphism between $V$ and $V^*$. (Fun fact: This is often called a "musical isomorphism" because in Einstein notation, the effect is to raise/lower, i.e., sharpen/flatten, indices.)
A choice of basis $e_i$ for $V$ also induces an isomorphism between $V$ and $V^*$ by mapping the basis to its dual basis, $e_i\mapsto e^i$, and then extending to all of $V$ by linearity.
The canonical basis of $\Bbb{R}^n$ gives an isomorphism between $V$ and $V^*$. The inner product also gives an isomorphism. In the case of Euclidean space, the two isomorphisms coincide on every tangent space (that is, the canonical basis is orthornormal with respect to the inner product). Teachers often do not make this distinction in elementary linear algebra classes. I would speculate this is for two reasons: One, it's a subtle point that may be lost on students without enough mathematical maturity; two, elementary linear algebra often focuses on matrix algebra computations instead of the structure of vector spaces.
Also, as Eric Wofsey points out, sometimes teachers do make a distinction by identifying one-forms with row vectors and vectors with column vectors. The musical isomorphisms are then given by transposition. I believe this is the case in Hubbard-Hubbard's book on linear algebra and multivariable calculus.

Beware conflating "equal" and "naturally isomorphic"! The two are not the same set or objects. There is an identification between them and they have the same properties. Think of different types of identifications as classification. In the same way, when speaking precisely, we don't say all mammals are equal, or the same, even though they have the same characteristic "mammalness" properties.
