Torsion is c infinity bilinear and skew symmetric

Expand $T(fX,Y)=f{\mathrm{\nabla }}_{X}Y-(Y(f)X+f{\mathrm{\nabla }}_{Y}X)-(f[X,Y]-Y(f)X)=fT(X,Y)$. Also $T(Y,X)=-T(X,Y)$ because the derivative terms exchange and the bracket is skew. This implies function-linearity in the second argument, while additivity and real linearity follow termwise.

Locally write $X=X^i{\mathrm{\partial }}_i$, $Y=Y^j{\mathrm{\partial }}_j$. Bilinearity gives $T(X,Y)={\sum }_{i,j}{X}^i{Y}^{j}T({\mathrm{\partial }}_i,{\mathrm{\partial }}_j)$; thus the value depends only on $X(p),Y(p)$. The displayed basis values are smooth by the smooth connection and bracket. This constructs the smooth fibrewise alternating bilinear map, whose values agree across frames since the original expression is intrinsic. Skew-symmetry gives $T(X,X)=0$ over the reals. In dimensions zero and one all alternating pairs vanish. Zero input gives zero, and the construction on an empty base is unique. No global frame or choice axiom is used.