Yeah, Schur did this a long time ago. Let $\tilde \Sigma_n \to \Sigma_n$ be a double cover (there are two). Schur uses the notation $[a_1 a_2 \cdots a_k]$ for a specific lift of the cycle $(a_1 a_2 \cdots a_k) \in \Sigma_n$. Then his presentation goes like this:
$$[a_1 a_2 \cdots a_k] = [a_1 a_2 \cdots a_i][a_i a_{i+1} \cdots a_k] \ \ \forall 1 < i< k$$
$$[a_1 a_2 \cdots a_k]^{[b_1 b_2 \cdots b_j]} = (-1)^{j-1}[\phi(a_1) \phi(a_2) \cdots \phi(a_k)]$$
where $\phi$ is the cycle $(b_1 b_2 \cdots b_j)$
$$[a_1 a_2 \cdots a_k]^k = \pm 1$$
according to which extension of $\Sigma_n$ you're interested in and:
$$[a_1 a_2 \cdots a_k][b_1 b_2 \cdots b_j] = (-1)^{(k-1)(j-1)}[b_1b_2 \cdots b_j][a_1 a_2 \cdots a_k]$$
provided the cycles $(a_1 a_2 \cdots a_k)$ and $(b_1 b_2 \cdots b_j)$ are disjoint.
The map $\tilde \Sigma_n \to \Sigma_n$ sends $[a_1 \cdots a_k]$ to $(a_1 \cdots a_k)$. So this gives you a corresponding presentation of the double of $A_n$.
A small extra tidbit -- if you think of $\Sigma_n$ as being the group of orientation-preserving isometries of $\mathbb R^{n}$ that preserves a regular (n-1)-simplex lifted to $Spin(n)$, the extension you want is the one where $[a_1 a_2 \cdots a_k]^k = -1$.