TITLE: Symmetric and anti-symmetric parts of the covariant derivative of a connection
QUESTION [9 upvotes]: The following is an excerpt from Sharpe's Differential Geometry - Cartan's Generalization of Klein's Erlangen Program.

Now we come to the question of higher derivatives. As usual in modern 
  differential geometry, we shall be concerned only with the skew-symmetric 
  part of the higher derivatives. In essence, what we shall be doing is taking 
  the partial derivatives with respect to the base (i.e., manifold) variables 
  and skew-symmetrizing the result, thus forgetting about the part of the 
  higher derivatives that vanish under this procedure. However, this will not 
  be made explicit in our treatment. The part of the higher derivative that 
  disappears has not been studied much in differential geometry since Elie 
  Cartan showed how useful it is to consider only the skew-symmetric part, 
  that is, the exterior derivative. The old masters did use the symmetric part...

"Partial derivatives with respect to the base" must be the covariant derivative of the connection. If I correctly understand what's written in this answer, then we have for any torsion free connection on a manifold the equality $ \mathrm dw=\operatorname{Alt}(\nabla w)$.
This already seems rather remarkable since the exterior derivative is intrinsic.
Question 1. What is the geometric meaning of the above fact for torsion-free connections? How does taking anti-symmetrization "forget" the structure of a horizontal bundle (connection)?
Question 2. Still for a torsion-free connection, what is the geometric meaning of "the part of the higher derivative that vanish under this procedure" of anti-symmetrization (i.e the symmetric part)?
Question 3. What's the conceptual picture for an arbitrary connection on a manifold (possibly with torsion)? Is the anti-symmetrization still the exterior derivative? What is the symmetric part?
Remark on Q3. I vaguely recall being told that torsion measures the failure of the fundamental theorem of calculus, and also that in torsion-free connections the parallel transport commutator is given by the Lie bracket (again, the latter is intrinsic).

REPLY [2 votes]: Although higher derivatives are best thought through with jet bundles where we actually differentiate a bundle and not just a manifold, there is a useful description of second derivatives using secondary tangent bundles, this is just iterating the tangent bundle, ie $TTM$. This is described in one of the early chapters in Kolár, Michor & Slováks book on natural bundles. (They also describe a generalisation of this in one of the later chapters which they call section forms - but they don't do anything with it).

Question 3. What's the conceptual picture for an arbitrary connection on a manifold (possibly with torsion)? Is the anti-symmetrization still the exterior derivative? What is the symmetric part?

On the last part of this question you may be interested in this paper, A Description of the Derivations of the Algebra of Symmetric Tensors, by A. Heydari, N. Boroojerdian, E. Peyghan and published in Archivum Mathematicum. They introduce symmetric forms, bracket, Lie derivative and differential. Whilst the symmetric differential is not nilpotent, ie $(d^s)^2$ does not vanish, they show

proposition 5:
Let $(M, g)$ be a Riemmanian manifold with Levi-Civita connection
$∇$. The 1-form $ω$ is Killing if and only if $d^sω = 0$.

and

proposition 7:
Every derivation of degree 1 on symmetric forms, whose value for a function is the differential of the function, is of the form of a symmetric differential of a connection which is also unique.