Is there geometric interpretation to Skew symmetric coefficient matrix, We know that the Frenet-Serret equation implies that the coefficient matrix of $\dot t,\dot n,\dot b$  is anti symmetric wrt $t,n,b$. But is there any geometric intuition that immediately gives this result? Thanks!
 A: Antisymmetric matrices describe linear maps involving oriented planes.  In this case, the plane is the one that describes the instantaneous rotation (and dilation*) of the frame.  The system rotation is described in two parts--a part the $tn$-plane, describing how the curve bends, and a part in the $nb$-plane, describing how the frame twists around the path of the curve.  Non-unit coefficients describe how the frame also dilates or shrinks in these planes.
(*) I say "dilation" here.  That is to say, the "natural" description of a frame of basis vectors is not necessarily a frame of unit vectors, and the coefficients reflect that.
A: If $v$ is a vector $= [v_x  v_y  v_z]^T$ in the frame described by versors $i, j, k$ then skew-symetric matrix $S(v)$ assigned to the vector $v$ is constructed as follows 
$$S(v) = \begin{pmatrix} v \times i \ \ v \times j \ \ v \times k \end{pmatrix}$$ (cross products of vector $v$ with versors)
From this construction geometric properties of skew-symmetric matrix follow, e.g. that columns of $S(v)$ are coplanar, vectors (columns) lie in the plane perpendicular to the vector $v$.
