The solution space of $x^{(n)}(t) = a_0 x(t) +a_1 x'(t) + \dots + a_{n-1} x^{(n-1)}(t)$ has dimension $n$.
Let's consider $n=2$: suppose we're given a homogeneous linear system consisting of three (scalar) DE of second order. Can I determine the exact dimesion of the solution space, or is it not possible without further information?
Yes it is, since every ODE of second order can be turned into a system of two ODE of order one, your system can be turned into an ODE system of 6 equations of order 1.
Assuming all your equations are independent (that it no equation can be obtained from the orders) the dimension of your solution space is 6