$R$ is determined by $\theta=(\partial_tR)R^T$, so $$R(t)=Texp(\int_0^t\theta(\tau)d\tau).$$
$R$ is a time-dependent orthogonal matrix. I don't understand how this expression is obtained, because when we time both side of the equation on the right with $R$, we get $\theta R=\partial_tR$, then integrate with respect to time, we get $R=\int_0^t\theta R(\tau)d\tau$. I don't understand how the exponential shows up and I can't find what $T$ is in the paper.