What is meant by "The Levi-Civita Connection is an $\mathfrak{so}(n)$-Valued 1-form"? this is a statement that I've seen around, and I thought it's time that I understand it. I know that the LCC is locally given by a matrix $ \omega = (\omega_i^j)$ of 1-forms in a preferred frame $e_i$, so that $$ \nabla f_ie_i  =   df_i \otimes e_i + f_i \omega_i^j \otimes e_j   $$ for any local smooth functions $f_i$. Then, $\omega$ is a matrix representing a linear map on each tangent space. 
Now, "$\nabla$ is an $\mathfrak{so}(n)$-valued 1-form" suggests to me that each $(\nabla v)|_p$ is in $\mathfrak{so}(T_pM)$ , but I know that this is only true for $v$ a Killing field. 
But perhaps I'm getting confused between $\nabla$ as an object and its representation $\omega$ in a particular frame. So, my next guess is that it means that, in some choices of local frame $e_i$, the matrix $\omega_i^j(v)$ is skew-symmetric for any $v$. Orthormal frame is the probable condition. But this would mean, in particular, that each $\nabla e_i$ is skew-symmetric, since if $v = e_i$, there are no nonconstant components of $v$ to worry about, and '$\nabla = \omega$'. Then again, we'd be at the statement that all the $e_i$ are (local) Killing fields, which is just rubbish - on a generic Riemannian manifold, there are no nontrivial local Killing fields, if I remember right.
So, what does "$\omega$ is $\mathfrak{so}$-valued" mean? Any help would be massively appreciated.

Generally a Lie-algebra valued $k$-form on $M$ is a smooth section of the bundle $(\mathfrak g\times M)\otimes \bigwedge^k T^*M$. In your case $\mathfrak g=\mathfrak {so}(n)$ and $k=1$.
See here
