Example 2 Given a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 2, connect $D_{1} A$, $D_{1} B$, $E$ is the midpoint of $D_{1} B$, $F$ is the midpoint of $B C$, $G$ is the midpoint of $A D_{1}$, find the size of the angle formed by the skew lines $D G$ and $E F$.