I have implemented a key pair generation scheme for RSA algorithm. I have taken the length of private key exponent as RSA key size, but then I've got to know that RSA key size is the size of the modulus $n$. Is it wrong to take RSA key size as the size of the private key exponent?
To generate a private key of 1024 bits, I increase the size of private key exponent regardless the modulus $n$.
No, the RSA key size is not the size of the private key exponent. It is customarily the number of bits in the public modulus (which is known as $N$). In other words, the key size is the integer $k$ such that $2^{k-1}\le N<2^k$.
In most implementations (and all implementations conforming to PKCS#1), a private exponent $d$ has size in bits at most the key size $k$, and typically is few bits smaller. Notice that the private exponent is not uniquely defined.
The smallest private exponent is $d=e^{-1}\bmod\lambda(N)$. It is always less than $\lambda(N)$, where $\lambda()$ is Carmichael's function, with $\lambda(N)=\operatorname{lcm}(p-1,q-1)$ when $N=p\cdot q$ with $p$ and $q$ distinct primes. The smallest private exponent always has size in bits strictly less than the key size $k$, and typically is a few bits smaller.
Sometime, a private exponent is computed as $d=e^{-1}\bmod\varphi(N)$, where $\varphi()$ is Euler's function, with $\varphi(N)=(p-1)\cdot(q-1)$ when $N=p\cdot q$ with $p$ and $q$ distinct primes. That particular $d$ has size in bits at most the key size $k$, often is a few bits smaller, and often is a few bits larger than the smallest private exponent.
Yet other methods to define a private exponent give no maximum for $d$, and only require that $d\equiv e^{-1}\pmod{\lambda(N)}$ (which is the necessary and sufficient condition for $d$ to work), or $d\equiv e^{-1}\pmod{\varphi(N)}$.
In addition to fgrieu's correct answer, I believe I want to emphasize something: increasing the size of the private exponent beyond the size of the modulus does absolutely nothing to improve security. If you want to increase the strength of the RSA key, you must increase the size of the moduus.
As fgrieu mentioned, for any $e$ there is a minimal $d = e^{-1} \bmod \lambda(N)$; there is nothing preventing you from using a larger $d$; that means extra work on your part, but any operation you do is precisely equivalent to the smaller $d$, and hence the extra work buys you absolutely nothing.
In addition, we commonly use the Chinese Remainder Thereom to optimize the RSA private operations; when doing this, we actually compute:
$$C_p = (P \bmod p) ^ {d\ \bmod\ p-1} \bmod p$$
$$C_q = (P \bmod q) ^ {d\ \bmod\ q-1} \bmod q$$
(and then combine $C_p$ and $C_q$ to form the actual ciphertext)
We work with $d\ \bmod\ p-1$ and $d\ \bmod\ q-1$; it turns out these values are identical for any "equivalent" private exponent, and so a larger-than-necessary private exponent doesn't actually do anything (either good or bad)
