The standard way of doing this is to regard the space $\Lambda$ of lines in  (or oriented lines) $\mathbb{R}^3$ as a homogeneous space of the group $G$ of Euclidean motions.  Then there is a natural $G$-invariant Riemannian metric on $\Lambda$, and you take the metric from that.  I could calculate that explicitly if you are having trouble with it.

Another method is to note that the space of oriented lines is naturally equivalent to the tangent bundle of $S^2$, namely, given a line $l$, you let $u(l)\in S^2$ be the oriented direction of the line, and you let $v(l)\in u(l)^\perp$ be the point on $l$ that is closest to the origin.  Then the mapping $l\to \bigl(u(l),v(l)\bigr)$ embeds the space of oriented lines in the space $S^2\times\mathbb{R}^3$ and you can take the induced metric, for example.