Let $M$ be a smooth manifold of dimension $n$ and suppose $U$ is a chart near a point $p$. Let $c^k_{ij}$ be a collection of constants. Is it possible to find a collection of functions $\alpha_{ij}: U \to \mathbb{R}$ such that at every point $x \in U$ the matrix $[\alpha_{ij}(x)]$ belongs to $SO(n)$, $\alpha_{ij}(p) = \delta^i_j$ and $(\partial_k(\alpha_{ij}))(p) = c^k_{ij}$.
Is there an easy way to see this?
The condition $\alpha(x) \in SO(n)$ can be written $\alpha^T \alpha=I$, or equivalently $$\sum_l \alpha_{il} \alpha_{jl} = \delta_{ij}.$$ Taking the derivative at $p$ and assuming $\alpha_{ij}(p) = \delta_{ij},$ $\partial_k\alpha_{ij}(p)=c^k_{ij}$ yields $$c^k_{ij}+c^k_{ji}= 0. \tag 1$$ Thus this equation holding is a necessary condition. If we let $c_{(k)}$ denote the matrix with coefficients $c^k_{ij},$ then this can simply be written $c_{(k)} + c_{(k)}^T = 0;$ i.e. this matrix should be antisymmetric for each $k.$
Geometrically, $(1)$ says that each $c_{(k)}$ is an element of the Lie algebra $\mathfrak{so}(n) = T_I SO(n),$ and thus $v \mapsto \sum_k c_{ij}^k v_k$ is a linear map from $T_p U$ to $T_I SO(n).$ Since any linear map between tangent spaces of manifolds can be extended to a local smooth map between the manifolds themselves, we find that the condition $(1)$ is also sufficient.