I don't know how much background knowledge you have, maybe all of this is known to you and you are looking for something else, but it's the first thing to come to my mind. I've tried to phrase the same statement in different ways.
The Lie algebra of skew-symmetric matrices is the Lie algebra corresponding to the Lie group of orthogonal matrices.
In other words, the space of skew matrices is the tangent space at the identity of the manifold of orthogonal matrices.
The space of skew matrices can in some sense be thought of as the infinitesimal version of orthogonal transformations.
I don't have time to get into why this is useful in any detail, but in many cases Lie algebras are significantly easier to handle and still give a great deal of information about the corresponding group.
A part of all this is the following observation. If $A$ is a skew matrix, then its exponential, $\exp(A)$, is an orthogonal matrix.
There are many books and lecture notes available on Lie groups and algebras. Some sources can be found in the answers to this question.