If $X_S (A) = AS$ defines a smooth vector field, where $A \in O(n)$ (matrices with $A^{-1} = A^{T}$) and $S$ belonging to the space of skew matrices (matrices with $S^{T} = -S$). How would I calculate the flow of $X_S$ here?

I'm using the definition of flow as one whose flow domain is all of $\mathbb{R} \times O(n)$, where the flow is the map $\theta : O(n) \times \mathbb{R} \to O(n)$ such that for all $s,t \in \mathbb{R}$ and $A \in O(n)$ the following holds: $$\theta(A,0) = A \space\ \space\ \text{and} \space\ \space\ \theta(\theta(A,t),s) = \theta(A, s+t)$$ I'm only familiar with finding flow via solving an ODE (or system of them). Need help understanding how this works with Lie Groups.


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