Clearly, not all second-/higher-order formulas can be written as a family of first order formula's. Otherwise we could write the induction axiom for arithmetic as a set of first order formula's and then there would be no non-standard model of the first-order theory of arithmetic.
But are there (known) conditions on (the syntactic structure of) a second-/higher-order formula, so that if(f) the conditions are satisfied, we CAN write it as (a family of) first-order formulas?