Gram matrix
A bilinear form is an abstract two-input machine, but to compute with it you want numbers. The Gram matrix is the bookkeeping device: pick a basis, then record the form's value on every pair of basis vectors. Once that finite table is in hand, the form's value on any two vectors follows by plugging in coordinates — the whole infinite form is captured by one finite matrix.
Given a basis e_1, …, e_n of V and a bilinear form B, the Gram matrix is G with entries G_ij = B(e_i, e_j). Then for vectors with coordinate columns x and y one has B(u, v) = x^T G y. The form is symmetric exactly when G is symmetric, alternating exactly when G is alternating, and nondegenerate exactly when G is invertible. For an inner product on a real space, G is symmetric positive definite.
Changing basis by an invertible matrix P replaces G with P^T G P — this is congruence, not similarity (P^T, not P^{-1}, appears). Hence the right notion of ‘same form in different coordinates’ is congruence of matrices, and the invariants of a form are precisely the congruence invariants of its Gram matrix: rank, discriminant up to squares, and over R the signature.
For B(x, y) = x_1 y_1 + x_1 y_2 + x_2 y_1 + 3 x_2 y_2 on R^2 the Gram matrix is [1, 1; 1, 3]. Its determinant 3 - 1 = 2 > 0 and leading entry 1 > 0, so the form is positive definite.
A 2×2 Gram matrix passing the positive-definiteness test.