Precisely the same considerations hold for forms. A function $\omega$ which assigns $\omega(x)\in\Lambda ^{p}(M_x)$ for each $x\in M$ is called a $p$-form on $M$. If $f:W\to\mathbf{R}^{n}$ is a coordinate system, then $f^{*}\omega$ is a $p$-form on $W$; we define $\omega$ to be differentiable if $f^{*}\omega$ is. A $p$-form $\omega$ on $M$ can be written as $$ \omega =\sum_{i_1<\cdots<i_p}\omega _{i_1,\ldots,i_p}\,\mathrm{d}x^{i_1}\wedge\cdots\wedge\,\mathrm{d}x^{i_p} .$$ Here the functions $\omega _{i_1,\ldots,i_p}$ are defined only on $M$. The definition of $\mathrm{d}\omega$ given previously would make no sense here, since $D_j(\omega _{i_1,\ldots,i_p})$ has no meaning. Nevertheless, there is a reasonable way of defining $\mathrm{d}\omega$.
Spivak says the definition of $\mathrm{d}\omega$ for a $k$-form $\omega$ does not make sense on a manifold because $D_j(\omega_{i_1, \dots , i_p})$ has no meaning. Does it have no meaning because the function $\omega_{i_1, \dots , i_p}$ is defined only on $M$ so that we have consider the tangent space but $\mathrm{d}\omega$ is taking partial derivatives of the function based on the unit vectors of $\mathbb{R}^n$ instead of the tangent space?