Definition (wedge product bundle):

Let ${\displaystyle M}$ be a differentiable manifold and ${\displaystyle \bigoplus ^{n}T^{*}M}$ be the ${\displaystyle (0,n)}$-tensor bundle of ${\displaystyle M}$. The wedge product bundle of degree ${\displaystyle n}$, denoted ${\displaystyle \bigwedge ^{n}T^{*}M}$, is the subbundle of ${\displaystyle \bigoplus ^{n}T^{*}M}$ made of the elements of ${\displaystyle \bigoplus ^{n}T^{*}M}$ which are invariant under all vector bundle isomorphisms

Definition (n-form):

Let ${\displaystyle M}$ be a differentiable manifold. An ${\displaystyle n}$-form on ${\displaystyle M}$ is an element of

${\displaystyle \Gamma \left(\bigwedge ^{n}T^{*}M\right)}$,

the sections of the vector bundle ${\displaystyle \bigwedge ^{n}T^{*}M}$.