Isomorphism between S(n) and a subgroup of A(n+2)

Show that S_n is isomorphic to a subgroup of A_n+2.

I will demonstrate my idea for n=3.

S₃ has 3!=6 permutations, 3 odd: (1 2), (1 3), (2 3); and 3 even: identity, (1 2 3)=(1 2)(2 3), (1 3 2)=(1 3)(3 2).

Let's consider them separately. Put elements 4 and 5 in A₅ aside. 

Identify even permutations in S₃ with permutations in A₅ with the same cycles as in S₃ while the elements 4 and 5 are fixed, e.g., (1 2)(2 3) in S₃ ↔ (1 2)(2 3) in A₅.

Identify odd permutations in S₃ with permutations in A₅ with the same cycles as in S₃ plus the cycle (4 5), e.g., (1 2) in S₃  ↔ (1 2)(4 5) in A₅.

I think, it is obvious how to generalize it for any n, right?