Skew-Symmetric Matrix:
A skew-symmetric matrix is a square matrix in which the transpose of the matrix is equal to the negation of the original matrix. In other words, for a skew-symmetric matrix A, the following condition holds: A^T = -A.

Eigenvalues:
Eigenvalues are the values λ for which the matrix equation A*v = λ*v has a non-zero solution v. In other words, they are the values that satisfy the equation Av = λv, where A is a square matrix and v is a non-zero vector.

Explanation:
The eigenvalues of a skew-symmetric matrix are either zero or purely imaginary. Let's understand why.

1. Skew-Symmetric Matrix Property:
For a skew-symmetric matrix A, the transpose of the matrix is equal to the negation of the original matrix.
A^T = -A

2. Eigenvalue Property:
For any matrix A, the eigenvalues satisfy the equation Av = λv, where A is the matrix, v is the eigenvector, and λ is the eigenvalue.

3. Derivation:
Let's assume v is an eigenvector of A with eigenvalue λ.
Av = λv

Taking the transpose of both sides:
(Av)^T = (λv)^T

v^T * A^T = λ * v^T

Since A is a skew-symmetric matrix (A^T = -A):
v^T * (-A) = λ * v^T

Multiplying both sides by -1:
-v^T * A = -λ * v^T

Comparing this equation with Av = λv, we can see that v^T is also an eigenvector of A with eigenvalue -λ.

4. Conclusion:
From the above derivation, we can conclude that if λ is an eigenvalue of a skew-symmetric matrix A, then -λ is also an eigenvalue. Since the eigenvalues are either zero or non-zero, the possibilities are:
- λ = 0 (zero eigenvalue)
- λ = ai (purely imaginary eigenvalue)

Hence, the correct answer is option 'C' - the eigenvalues of a skew-symmetric matrix are either zero or purely imaginary.