Skew lines and what's between them Is it always possible to find a line perpendicular to two skew lines in space?
And how can we visualise the proof geometrically? And if anyone could present the proof that it is always possible to exist a line perpendicular to both skew lines, please elaborate.

You can do better than to just arrive at the separation distance of two skew lines at closest approach.  With some software computing, you can solve for the specific line segment bridging between them (located by its endpoints):
This approach takes advantage of extrusion.
We can obtain the 3D run direction of the bridging line segment LS by taking the normalized cross-product of the run-dirs of the two lines:
dir-LS <-- (dir-L1 X dir-L2) norm    (mutual orthogonal direction to d1 and d2)
Using dir-LS, extrude L1 into a plane PL1.  All you need to do is generate 2 points on L1, L1p1 and L1p2, and then obtain a 3rd point p3 = L1p1 + dir-LS. These 3 points spawn the plane PL1 extruding L1 in direction LS-dir:
PL1 <-- extrude( L1, LS-dir )
The point where L2 pierces through PL1 gives endPt2 of the bridge LS:
LS.endPt2 <-- intersectionOf ( L2, PL1 )
By symmetry, we can compute the location of LS.endPt1:
PL2 <-- extrude( L2, LS-dir )
The point where L1 pierces through PL2 gives endPt1 of the bridge LS:
LS.endPt1 <-- intersectionOf ( L1, PL2 )
With both endpoints, we've solved for the bridging line segment of the 2 skew lines.  (The length of LS gives the separation distance.)
