Let $g_t : M \rightarrow M$ be the flow of a vector field, that is a family of diffeomorphisms such that $g_t \circ g_s = g_{t+s}$ to $t,s > 0$ and $g_0 = \operatorname{id}$.
One can interpret the diffeomorphism group as an infinite dimensional Lie group, up to some technicalities I am not interested in, with its Lie algebra given by vector fields. This group admits an exponential map that sometimes coincides with the Riemannian exponential map, namely there is an $\exp: \mathfrak{X}(M) \rightarrow Diff(M)$ such that:
$$\exp(tX) = g_t$$
Where $X : M \rightarrow TM$ is a vector field satisfying
$$\frac{d}{dt}g_t(x) = X(g_t(x))$$
My question is, using a coordinate chart a diffeomorphism can be represented locally as just a collection of mappings $g = (g_t^i)_{i=1}^n$ with $g^i_t = F^i(x,t) : \mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R}^n$. Similarly, a vector field can be identified as a first order differential operator that has components with respect to a basis induced by the same chart.
Can we say, that in a small neighbourhood we have:
$$\exp(tX) = \sum \frac{1}{k!}X^k$$
Where the RHS is some sort of operator/matrix exponential instead of some abstract time 1 geodesic map?