Indiscernible to create descending chain of elementary models 
Let $M$ an infinite structure such that $\mid M \mid \ge \mid L(M) \mid $. Show that exists a proper elementary extension $N$ and a chain $\langle N_{i} \mid i < \omega \rangle $ such that $$N=N_{0},    \  \ N_{i} \succ N_{i+1} $$ for all $i< \omega $ and $M=\bigcap_{i< \omega} N_{i}$

That this exercise comes as an application of sequence of indiscernible, the procedure would be to find a sequence of indiscernible and with this sequence and with $M$  build the first model $N_{0}$ of the chain of elementary extensions of $M$. I'm not sure if that is the correct. Thanks for any hint.

Expand the language with Skolem functions. Let $\{a_i: i<\omega\}$ be some elements such that $a_n\notin\operatorname{dcl}(\{a_i: n<i<\omega\}\cup M)$. (The definability closure is relative to the Skolemized language.) Any indiscernible sequence certainly meets this requirement. Let $N_n$ be (the reduct to $L$ of) the structure generated by $M\cup\{a_i: n<i<\omega\}$.
Question. Is there an answer that does not use Skolem functions?  
