Consider a connection $\nabla$ on the tangent bundle $TM$ some manifold $M$. For a such a conneciton the torsion is defined as $T(X,Y)=\nabla_X Y -\nabla_Y X -[X,Y]$.
I want to show that $T\in \Omega^2(M,TM)$. That is to show that $T$ is a $2$ form that takes values in the tangent bundle.
It's clearly anti-symmetric in $X$ and $Y$. We also must show that it is multi linear but this is also clear.
Individually $\nabla_X Y$,$\nabla_Y X$, and $[X,Y]$ take values in $TM$ but why does there sum also take values? Is this because, at any point $p$ in $M$ $\nabla_X Y$,$\nabla_Y X$, and $[X,Y]$ will all take values in $T_pM$ which is a vector space?
Yes, that's right. But the crucial thing to show you have a well-defined tensor field is that it's multilinear not over the scalars but over the $C^\infty$ functions. That's why the Lie bracket term has to be there.