Given only one copy of such a state, it is not possible to determine it with any good probability. Reason being there is no way, in principle, to extract information from the system without making a measurement on the system. And when we go for a measurement, we project it to a basis element $i,j$ if we turn out to pick these by chance from the set $\{1,2,...,N\}$, in otherwise cases, we can never make out what is $i$ or $j$ from the given set of degrees of freedom. To answer a more general question, if we don't even know what the set $\{1,2,...,N\}$ is; Suppose someone prepares a quantum state and gives to you without telling what kind of experimental apparatus may be the basis, there is no way in principle to figure out anything (if someone gives $n$ isolated levels of a harmonic oscillator and you are looking for two states of polarization in it, for instance). Hence, in all literature, we assume that the type of the experimental apparatus is always known, which means that the basis $\{1,2,...,N\}$ is known but a general state $|\psi\rangle=\frac{1}{\sqrt{N}}\sum_i c_i |i\rangle$ is unknown in this Basis, where we can evaluate the coefficients $c_i$s by repeated measurements. One such technique is the **State Tomography**. Otherwise, the most ordinary and default way is to use the projectors $|i\rangle \langle i|$ formed from the basis and see what we get as the final state after the measurement.