Another conceptually nice definition of the exterior derivative is given in Bourbaki (Varietes differentielles et analytiques, Fascicule de resultats), (8.3.4) and (8.3.5). The idea is the following: if w$\omega$ is an exterior p$p$-form on X$X$, consider it as a section w: X to Omega^p(X)$\omega: X \to \Omega^p(X)$ of the bundle Omega^p(X)$\Omega^p(X)$ of p$p$-forms. It makes sense to take its derivative dw$d\omega$ at each point x in X$x \in X$. Then one sees that dw$d\omega$ corresponds to a p+1$p+1$ exterior form.
By the way, a natural and simple definition of tangent vector on a smooth manifold is given in the same book in (5.5.1).
