3. In a regular tetrahedron $A B C D$ with edge length 1, $M$ and $N$ are the midpoints of $A D$ and $B C$ respectively. Then the distance between the skew lines $A B$ and $M N$ is ( ).
(A) $\frac{1}{4}$
(B) $\frac{1}{12}$
(C) $\frac{\sqrt{2}}{4}$
(D) $\frac{\sqrt{3}}{8}$
3.C.

As shown in Figure 1, take the midpoints $E$ and $F$ of $AB$ and $CD$ respectively, and connect $EN$, $NF$, $FM$, $ME$, $EF$, $FB$, $FA$, and the intersection of $EF$ and $MN$ at point $O$. Clearly, quadrilateral $ENFM$ is a planar quadrilateral and is a rhombus, so $MN \perp EF$. Also, $EF$ is the median of isosceles $\triangle FAB$, thus $EF \perp AB$, and $EO = \frac{1}{2} EF$. Therefore, $EF = \frac{\sqrt{2}}{2}$.