Prove that every covariant tensor field $\o$ of order $1$ is of the form $$ \o=\sum_{j=1}^n f_j\,dx_j $$ for some smooth functions $f_j\in C^\infty(M)$.

Let $E^j$ be the vectorfield with components $(0,\ldots,0,1,0,\ldots,0)\in\R^n$ with $1$ in the $j$-th slot. Then any vector field $X$ has a unique expansion: $$ X=\sum_{j=1}^n\z_j\,E^j $$ for some smooth functions $\z_j:M\rar\R$ - these are just the components of $X$. By the definition of a tensor field $\o$ of order $1$: $$ \o(X) =\o(\sum\z_j\,E^j) =\sum_j\z_j\,\o(E^j) =\sum_j\o(E^j)\,dx_j(X) $$ and thus the conclusion follows with $f_j\colon=\o(E^j)$.