Skew Young tableaux are simple combinatorial objects arising in the theory of symmetric functions and the representation theory of the symmetric group. The number $f^\sigma$ of (standard) skew Young tableaux of skew shape $\sigma$ has a simple determinantal formula due to Aitken. We will discuss some situations for which there exist other formulas or generating functions for $f^\sigma$. For instance, for certain skew shapes $\sigma$ the number $f^\sigma$ can be described in terms of Euler numbers (the number of alternating permutations of $1,2,\dots,n$) using an analytic technique introduced by Elkies and further developed by Baryshnikov and Romik. Certain other sequences $\sigma_n$ of skew shapes have simple generating functions for the numbers $f^{\sigma_n}$, based on a well-known connection between determinants and generating functions.