$\newcommand{\Aut}{\mathrm{Aut}}$

The approach shown here is inspired by the writings of Eschenburg, Heintze, Jost, Karcher.

Let $E$ be a vector bundle over a smooth manifold $M$ and $\nabla$ a connection on $E$. The curvature of the connection is the section $\Omega$ of $\bigwedge^2T^*M\otimes \Aut(E)$ such that $$\label{curvature} \Omega(X,Y)e = ([\nabla_X,\nabla_Y] - \nabla_{[X,Y]})e \in E_x,$$ for any $x \in M$, $X, Y \in T_xM$, $e \in E_x$.

Given a smooth curve $c: [0,1] \rightarrow M$, the parallel transport of $e \in E_{c(0)}$ along $c$ is defined to be the section $f: [0,1] \rightarrow E$ such that the following hold for each $t \in [0,1]$: \begin{align*} f(t) &\in E_{c(t)}\\ f(0) &= e\\ \nabla_Tf(t) &= 0, \end{align*} where $T = \partial_t$. Denote $P_ce = f(1)$.

Let $c: [0,1] \rightarrow M$ be a $C^1$ null-homotopic curve based at $x$. There exists a $C^1$ map $C: [0,1]\times[0,1]\rightarrow M$ satisfying the following for each $0 \le s, t \le 1$: \begin{align*} C(0,t) &= x\\ C(1,t) &= c(t)\\ C(s,0) &= x\\ C(s,1) &= x. \end{align*}

Given $e_x \in E_x$, let $e: [0,1]\times [0,1] \rightarrow E$ be $C^2$ section of $C^*E$ satisfying the following for all $0 \le s, t \le 1$: \begin{align*} e(s,t) &\in E_{C(s,t)}\\ e(s,0) &= e_x\\ \nabla_Te(1,t) &= 0\\ \nabla_Se(s,t) &= 0, \end{align*} where $S = \partial_s$ and $T = \partial_t$. In particular, \begin{align*} e(s,1) &= P_ce_x. \end{align*}

Let $E^*$ be the dual vector bundle of $E$. Given $\varepsilon_x \in E^*_x$, let $\varepsilon: [0,1]\times[0,1] \rightarrow E^*$ satisfy the following for all $0 \le s, t \le 1$: \begin{align*} \varepsilon(s,t) &\in E^*_{C(s,t)}\\ \varepsilon(0,t) &= \varepsilon_x\\ \varepsilon(s,0) &= \varepsilon_x\\ \varepsilon(s,1) &= \varepsilon_x\\ \nabla_S\varepsilon(s,t) &= 0. \end{align*} It follows that \begin{align*} \nabla_T\varepsilon(0,t) &= 0. \end{align*}

$\langle\varepsilon_x, P_ce_x - e_x\rangle = \int_{[0,1]\times[0,1]} \langle\varepsilon(s,t),C^*\Omega e\rangle.$
\begin{align*} \langle \varepsilon_x, P_ce_x - e_x\rangle &= \langle \varepsilon(0,1), e(0,1)\rangle - \langle\varepsilon(0,0), e(0,0)\rangle\\ &= \int_{t=0}^{t=1} \partial_t(\langle\varepsilon(0,t),e(0,t)\rangle)\,dt\\ &= \int_{t=0}^{t=1} \langle\varepsilon,\nabla_Te(0,t)\rangle\,dt\\ &= \int_{t=0}^{t=1} \left[\langle\varepsilon,\nabla_Te(1,t)\rangle - \int_{s=0}^{s=1}\partial_s(\langle\varepsilon, \nabla_Te(s,t)\rangle)\,ds\right]\,dt\\ &= - \int_{t=0}^{t=1}\int_{s=0}^{s=1} \langle\varepsilon, \nabla_S\nabla_Te(s,t)\rangle\,ds\,dt\\ &= \int_{t=0}^{t=1}\int_{s=0}^{s=1} \langle\varepsilon, \Omega(C_*T,C_*S)e(s,t)\rangle\,ds\,dt\\ &= \int_{[0,1]\times[0,1]} \langle\varepsilon,C^*\Omega e\rangle . \end{align*}

A corollary of this is the Ambrose-Singer theorem. An elegant presentation of the above can be found in lecture notes of Werner Ballman.