This is known as Differential Galois Theory, first developed by Picard and Vessiot. In your case you should look for authors such as Kolchin or Singer and Van Der Put. Some systems definitely admit solutions in "closed form" (you can build them!), but most won't.
The ingredient is the "monodromy group", measuring the multivaluedness of the analytic continuation (in the complex line) of local solutions, whose Zariski closure is an algebraic subgroup of $GL(n,\mathbb C)$ (assuming your equation starts with coefficient in $\mathbb R$ or $\mathbb C$). The solvability of the (connected component of the identity) of this algebraic group is the criterion for solvability in "closed form".

A comprehensive reference is:

van der Put, Marius; Singer, Michael F

"Galois theory of linear differential equations"

Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 328. Springer-Verlag, Berlin, 2003.

MathSciNet : MR1960772