If M is a square matrix then the
The matrix exponential
The matrix exponential function
Let M be the 2by2 matrix 0, minus theta, theta, 0. We'll compute \exp(M). First let's compute the powers of M:

M squared equals minus theta squared, 0, 0 minus theta squared, which is just minus theta squared times the identity matrix,

M cubed equals 0, theta cubed, minus theta cubed, 0,

M to the 4 equals (M squared) squared, which is theta to the 4 times the identity.

M to the 5 equals theta to the 4 times M, etc
so we end up with: exp M equals the 2by2 matrix whose entries are: 1 minus a half theta squared plus one over 4 factorial theta to the 4 dot dot dot, minus theta plus one over three factorial theta cubed minus dot dot dot, theta minus one over three factorial theta cubed plus dot dot dot, 1 minus a half theta squared plus one over 4 factorial theta to the four dot dot dot, which is just the 2by2 matrix cos theta, minus sine theta, sine theta, cos theta where we have just observed that the power series in each entries are the Taylor series of cos and sin.
This is the 2by2 matrix that gives you a rotation by an angle \theta. You should imagine that, as in this example, the exponential map "eats" a very simple matrix (like an nbyn antisymmetric matrix) and outputs a much more complicated and useful matrix (like an nbyn rotation matrix).
Preclass exercise
Compute exp of the 2by2 matrix 0, x, 0, 0 and exp of the 3by3 matrix 0, a, c, 0, 0, b, 0, 0, 0.