This is an introductory question. I am bit confused how to write the Euclidean metric for example $$ds^2=dx^2+dy^2$$ in term of the wedge product?
This is two form and hence can be written in term of wedge of two one forms. One way to write using the outer product $$g_{\mu\nu}(dx^\mu \otimes dx^\nu)$$ But since $\wedge$ and $\otimes$ are related I should be able to write in term of the $\wedge$ but $$dx\wedge dx=0$$
This question is really about linear algebra, since the issue is visible separately on each tangent space.
It follows from the definition that linear combinations of wedge products are alternating tensors: $$(\alpha \wedge \beta)(X, Y) = \alpha(X) \beta(Y) - \beta(x) \alpha(Y) = -(\alpha \wedge \beta)(Y, X),$$ so for any $2$-form $\omega := \sum_i \alpha_i \wedge \beta_i$, $$\omega(X, Y) = -\omega(Y, X).$$ In particular, for any $X$, $$\omega(X, X) = 0 .$$
On the other hand, a Riemannian metric is a symmetric tensor: $$g(X, Y) = g(Y, X) .$$ So, if $g$ were a linear combination of wedge products, we would have for all $X, Y$ that $g(X, Y) = -g(Y, X) = -g(X, Y)$, hence $g = 0$, a contradiction (unless the manifold has dimension $0$).