Skew-Symmetric Matrix
A skew-symmetric matrix is a square matrix where the transpose of the matrix is equal to the negative of the matrix itself. In other words, if A is a skew-symmetric matrix, then A^T = -A.

Trace of a Matrix
The trace of a matrix is defined as the sum of the elements on the main diagonal of the matrix. For an n x n matrix A, the trace is given by Tr(A) = a_11 + a_22 + ... + a_nn, where a_ij represents the element in the i-th row and j-th column of the matrix.

Proof
To prove that the trace of a skew-symmetric matrix is zero, let's consider a skew-symmetric matrix A.

A = [a_11 a_12 a_13
a_21 a_22 a_23
a_31 a_32 a_33]

Taking the transpose of A, we have:

A^T = [a_11 a_21 a_31
a_12 a_22 a_32
a_13 a_23 a_33]

According to the definition of a skew-symmetric matrix, A^T = -A. So, we can write:

-a_11 = a_11
-a_21 = a_12
-a_31 = a_13
-a_12 = a_21
-a_22 = a_22
-a_32 = a_23
-a_13 = a_31
-a_23 = a_32
-a_33 = a_33

From these equations, we can see that the diagonal elements of A must be zero (a_11 = 0, a_22 = 0, a_33 = 0).

Therefore, the trace of A, Tr(A) = a_11 + a_22 + a_33 = 0 + 0 + 0 = 0.

Conclusion
The trace of a skew-symmetric matrix is always zero. Therefore, the correct answer is option c) 0.